1.2 algebraic expressions

Expressions

We order pizzas from Pizza Grande. Expressions

We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge.

Expressions

We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50,

Expressions

We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip.

Expressions

We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip.If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”.

Expressions

We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip.If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”.Such a formula is called an expression.

Expressions

We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip.If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”.Such a formula is called an expression.

Expressions

If we ordered x = 100 pizzas, the cost would be 8(100)+10 = $810.

We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip.If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”.Such a formula is called an expression.

Expressions

If we ordered x = 100 pizzas, the cost would be 8(100)+10 = $810. The value x = 100 is called the input and the projected cost $810 is called the output.

We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip.If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”.Such a formula is called an expression.

Expressions

Definition: Mathematical expressions are calculation procedures which are written with numbers, variables, operation symbols +, –, *, / and ( )’s.

If we ordered x = 100 pizzas, the cost would be 8(100)+10 = $810. The value x = 100 is called the input and the projected cost $810 is called the output.

We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip.If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”.Such a formula is called an expression.

Expressions

Definition: Mathematical expressions are calculation procedures which are written with numbers, variables, operation symbols +, –, *, / and ( )’s. Expressions calculate future results.

If we ordered x = 100 pizzas, the cost would be 8(100)+10 = $810. The value x = 100 is called the input and the projected cost $810 is called the output.

Algebraic Expressions

An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots.

Algebraic Expressions

An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots.

Algebraic Expressions

Examples of algebraic expressions are

3x2 – 2x + 4,

An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots.

Algebraic Expressions

Examples of algebraic expressions are

3x2 – 2x + 4,x2 + 3

3 x3 – 2x – 4 ,

An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots.

Algebraic Expressions

Examples of algebraic expressions are

3x2 – 2x + 4,x2 + 3

3 x3 – 2x – 4 ,

(x1/2 + y)1/3

(4y2 – (x + 4)1/2)1/4

An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots.

Algebraic Expressions

Examples of algebraic expressions are

3x2 – 2x + 4,x2 + 3

3 x3 – 2x – 4 ,

(x1/2 + y)1/3

(4y2 – (x + 4)1/2)1/4

Examples of non-algebraic expressions aresin(x), 2x, log(x + 1).

An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots.

Algebraic Expressions

Examples of algebraic expressions are

3x2 – 2x + 4,x2 + 3

3 x3 – 2x – 4 ,

(x1/2 + y)1/3

(4y2 – (x + 4)1/2)1/4

Examples of non-algebraic expressions aresin(x), 2x, log(x + 1).

The algebraic expressions anxn + an-1xn-1...+ a1x + a0 where ai are numbers, are called polynomials (in x).

An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots.

Algebraic Expressions

Examples of algebraic expressions are

3x2 – 2x + 4,x2 + 3

3 x3 – 2x – 4 ,

(x1/2 + y)1/3

(4y2 – (x + 4)1/2)1/4

Examples of non-algebraic expressions aresin(x), 2x, log(x + 1).

The algebraic expressions anxn + an-1xn-1...+ a1x + a0 where ai are numbers, are called polynomials (in x).

The algebraic expressions where P and Q are polynomials, are called rational expressions.

PQ

Polynomial ExpressionsFollowing are examples of operations with polynomials and rational expressions.

Polynomial ExpressionsFollowing are examples of operations with polynomials and rational expressions.

Example A. Expand and simplify.(2x – 5)(x +3) – (3x – 4)(x + 5)

Polynomial ExpressionsFollowing are examples of operations with polynomials and rational expressions.

Example A. Expand and simplify.(2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert [ ]

Polynomial ExpressionsFollowing are examples of operations with polynomials and rational expressions.

Example A. Expand and simplify.(2x – 5)(x +3) – [(3x – 4)(x + 5)]= 2x2 + x – 15 – [3x2 + 11x – 20]

Insert [ ]

Polynomial ExpressionsFollowing are examples of operations with polynomials and rational expressions.

Example A. Expand and simplify.(2x – 5)(x +3) – [(3x – 4)(x + 5)]= 2x2 + x – 15 – [3x2 + 11x – 20]= 2x2 + x – 15 – 3x2 – 11x + 20

Insert [ ]

Polynomial ExpressionsFollowing are examples of operations with polynomials and rational expressions.

Example A. Expand and simplify.(2x – 5)(x +3) – [(3x – 4)(x + 5)]= 2x2 + x – 15 – [3x2 + 11x – 20]= 2x2 + x – 15 – 3x2 – 11x + 20= –x2 – 10x + 5

Insert [ ]

Polynomial ExpressionsFollowing are examples of operations with polynomials and rational expressions.

Example A. Expand and simplify.(2x – 5)(x +3) – [(3x – 4)(x + 5)]= 2x2 + x – 15 – [3x2 + 11x – 20]= 2x2 + x – 15 – 3x2 – 11x + 20= –x2 – 10x + 5

Insert [ ]

To factor an expression means to write it as a product in a nontrivial way.

Polynomial ExpressionsFollowing are examples of operations with polynomials and rational expressions.

Example A. Expand and simplify.(2x – 5)(x +3) – [(3x – 4)(x + 5)]= 2x2 + x – 15 – [3x2 + 11x – 20]= 2x2 + x – 15 – 3x2 – 11x + 20= –x2 – 10x + 5

Insert [ ]

A3 B3 = (A B)(A2 AB + B2)

Important Factoring Formulas:

To factor an expression means to write it as a product in a nontrivial way.

A2 – B2 = (A + B)(A – B)+– +– +–

Example B. Factor 64x3 + 125Polynomial Expressions

Example B. Factor 64x3 + 12564x3 + 125 = (4x)3 + (5)3

Polynomial Expressions

Example B. Factor 64x3 + 12564x3 + 125 = (4x)3 + (5)3

= (4x + 5)((4x)2 – (4x)(5) +(5)2)

Polynomial Expressions

Example B. Factor 64x3 + 12564x3 + 125 = (4x)3 + (5)3

= (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25)

Polynomial Expressions

Example B. Factor 64x3 + 12564x3 + 125 = (4x)3 + (5)3

= (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25)

Polynomial Expressions

We factor polynomials for the following purposes.

Example B. Factor 64x3 + 12564x3 + 125 = (4x)3 + (5)3

= (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25)

Polynomial Expressions

We factor polynomials for the following purposes.

I. It’s easier to calculate an output or to check the sign of an output using the factored form.

Example B. Factor 64x3 + 12564x3 + 125 = (4x)3 + (5)3

= (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25)

Polynomial Expressions

We factor polynomials for the following purposes.

I. It’s easier to calculate an output or to check the sign of an output using the factored form.II. To simplify or perform algebraic operations with rational expressions.

Example B. Factor 64x3 + 12564x3 + 125 = (4x)3 + (5)3

= (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25)

Polynomial Expressions

We factor polynomials for the following purposes.

I. It’s easier to calculate an output or to check the sign of an output using the factored form.II. To simplify or perform algebraic operations with rational expressions. III. To solve equations (See next section).

Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions.

Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.

Example C. Evaluate 2x3 – 5x2 + 2x for x = -2, -1, 3 by factoring it first.

Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.

Example C. Evaluate 2x3 – 5x2 + 2x for x = -2, -1, 3 by factoring it first.2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2)

Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.

Example C. Evaluate 2x3 – 5x2 + 2x for x = -2, -1, 3 by factoring it first.2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2)Plug in x = -2:-2 [2(-2) – 1] [(-2) – 2]

Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.

Example C. Evaluate 2x3 – 5x2 + 2x for x = -2, -1, 3 by factoring it first.2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2)Plug in x = -2:-2 [2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40

Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.

Example C. Evaluate 2x3 – 5x2 + 2x for x = -2, -1, 3 by factoring it first.2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2)Plug in x = -2:-2 [2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40 Plug in x = -1:-1 [2(-1) – 1] [(-1) – 2]

Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.

Example C. Evaluate 2x3 – 5x2 + 2x for x = -2, -1, 3 by factoring it first.2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2)Plug in x = -2:-2 [2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40 Plug in x = -1:-1 [2(-1) – 1] [(-1) – 2] = -1 [-3] [-3] = -9

Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.

Example C. Evaluate 2x3 – 5x2 + 2x for x = -2, -1, 3 by factoring it first.2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2)Plug in x = -2:-2 [2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40 Plug in x = -1:-1 [2(-1) – 1] [(-1) – 2] = -1 [-3] [-3] = -9 Plug in x = 3:3 [2(3) – 1] [(3) – 2]

Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.

Example C. Evaluate 2x3 – 5x2 + 2x for x = -2, -1, 3 by factoring it first.2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2)Plug in x = -2:-2 [2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40 Plug in x = -1:-1 [2(-1) – 1] [(-1) – 2] = -1 [-3] [-3] = -9 Plug in x = 3:3 [2(3) – 1] [(3) – 2] = 3 [5] [1] = 15

Evaluate Polynomial ExpressionsIt's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.

Determine the Signs of the Outputs.It's easier to determine the sign of an output, when evaluating an expression, using the factored form.

Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2.

Determine the Signs of the Outputs.It's easier to determine the sign of an output, when evaluating an expression, using the factored form.

Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2.x2 – 2x – 3 = (x – 3)(x + 1).

Determine the Signs of the Outputs.It's easier to determine the sign of an output, when evaluating an expression, using the factored form.

Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2.x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = -3/2,we get (-3/2 – 3)(-3/2 + 1)

Determine the Signs of the Outputs.It's easier to determine the sign of an output, when evaluating an expression, using the factored form.

Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2.x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = -3/2,we get (-3/2 – 3)(-3/2 + 1) is (–)(–) = + .

Determine the Signs of the Outputs.It's easier to determine the sign of an output, when evaluating an expression, using the factored form.

Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2.x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = -3/2,we get (-3/2 – 3)(-3/2 + 1) is (–)(–) = + .

Determine the Signs of the Outputs.

Rational ExpressionsWe say a rational expression is in the factored formif it's numerator and denominator are factored.

It's easier to determine the sign of an output, when evaluating an expression, using the factored form.

Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2.x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = -3/2,we get (-3/2 – 3)(-3/2 + 1) is (–)(–) = + .

Determine the Signs of the Outputs.

Rational ExpressionsWe say a rational expression is in the factored formif it's numerator and denominator are factored.

Example E. Factor

x2 – 1 x2 – 3x+ 2

It's easier to determine the sign of an output, when evaluating an expression, using the factored form.

Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2.x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = -3/2,we get (-3/2 – 3)(-3/2 + 1) is (–)(–) = + .

Determine the Signs of the Outputs.

Rational ExpressionsWe say a rational expression is in the factored formif it's numerator and denominator are factored.

Example E. Factor

x2 – 1 x2 – 3x+ 2

x2 – 1 x2 – 3x+ 2

= (x – 1)(x + 1) (x – 1)(x – 2)

is the factored form.

It's easier to determine the sign of an output, when evaluating an expression, using the factored form.

Rational ExpressionsWe put rational expressions in the factored form in order to reduce, multiply or divide them.

Rational ExpressionsWe put rational expressions in the factored form in order to reduce, multiply or divide them.

Rational ExpressionsWe put rational expressions in the factored form in order to reduce, multiply or divide them. Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled,

i.e. x*yx*z = x*y

x*z = yz

1

Rational ExpressionsWe put rational expressions in the factored form in order to reduce, multiply or divide them.

x*yx*z = x*y

x*z = yz

A rational expression that can't be cancelled any further is said to be reduced.

Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled,

i.e.

Rational ExpressionsWe put rational expressions in the factored form in order to reduce, multiply or divide them.

Example F. Reduce x2 – 1 x2 – 3x+ 2

x*yx*z = x*y

x*z = yz

A rational expression that can't be cancelled any further is said to be reduced.

Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled,

i.e.

Rational ExpressionsWe put rational expressions in the factored form in order to reduce, multiply or divide them.

Example F. Reduce x2 – 1 x2 – 3x+ 2

x2 – 1 x2 – 3x+ 2 = (x – 1)(x + 1)

(x – 1)(x – 2)

x*yx*z = x*y

x*z = yz

A rational expression that can't be cancelled any further is said to be reduced.

factor

Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled,

i.e.

Rational ExpressionsWe put rational expressions in the factored form in order to reduce, multiply or divide them.

Example F. Reduce x2 – 1 x2 – 3x+ 2

x2 – 1 x2 – 3x+ 2 = (x – 1)(x + 1)

(x – 1)(x – 2)

x*yx*z = x*y

x*z = yz

A rational expression that can't be cancelled any further is said to be reduced.

= (x + 1) (x – 2)

factor

Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled,

i.e.

Rational ExpressionsMultiplication Rule: PQ

RS* = P*R

Q*S

Rational ExpressionsMultiplication Rule: PQ

RS* = P*R

Q*S

Division Rule: PQ

RS

÷ = P*SQ*R

Reciprocate

Rational ExpressionsMultiplication Rule:

To carry out these operations, put the expressions in factored form and cancel as much as possible.

PQ

RS* = P*R

Q*S

Division Rule: PQ

RS

÷ = P*SQ*R

Reciprocate

Rational ExpressionsMultiplication Rule:

To carry out these operations, put the expressions in factored form and cancel as much as possible.

PQ

RS* = P*R

Q*S

Division Rule: PQ

RS

÷ = P*SQ*R

Reciprocate

Example G. Simplify

(2x – 6) (y + 3) ÷ (y2 + 2y – 3)

(9 – x2)

Rational ExpressionsMultiplication Rule:

To carry out these operations, put the expressions in factored form and cancel as much as possible.

PQ

RS* = P*R

Q*S

Division Rule: PQ

RS

÷ = P*SQ*R

Reciprocate

Example G. Simplify

(2x – 6) (y + 3) ÷ (y2 + 2y – 3)

(9 – x2)

(2x – 6) (y + 3) ÷ (y2 + 2y – 3)

(9 – x2) = (2x – 6) (y + 3)

(y2 + 2y – 3) (9 – x2)*

Rational ExpressionsMultiplication Rule:

To carry out these operations, put the expressions in factored form and cancel as much as possible.

PQ

RS* = P*R

Q*S

Division Rule: PQ

RS

÷ = P*SQ*R

Reciprocate

Example G. Simplify

(2x – 6) (y + 3) ÷ (y2 + 2y – 3)

(9 – x2)

(2x – 6) (y + 3) ÷ (y2 + 2y – 3)

(9 – x2) = (2x – 6) (y + 3)

(y2 + 2y – 3) (9 – x2)*

= 2(x – 3) (y + 3)

(y + 3)(y – 1) (3 – x)(3 + x)*

Rational ExpressionsMultiplication Rule:

To carry out these operations, put the expressions in factored form and cancel as much as possible.

PQ

RS* = P*R

Q*S

Division Rule: PQ

RS

÷ = P*SQ*R

Reciprocate

Example G. Simplify

(2x – 6) (y + 3) ÷ (y2 + 2y – 3)

(9 – x2)

(2x – 6) (y + 3) ÷ (y2 + 2y – 3)

(9 – x2) = (2x – 6) (y + 3)

(y2 + 2y – 3) (9 – x2)*

= 2(x – 3) (y + 3)

(y + 3)(y – 1) (3 – x)(3 + x)*

1

Rational ExpressionsMultiplication Rule:

To carry out these operations, put the expressions in factored form and cancel as much as possible.

PQ

RS* = P*R

Q*S

Division Rule: PQ

RS

÷ = P*SQ*R

Reciprocate

Example G. Simplify

(2x – 6) (y + 3) ÷ (y2 + 2y – 3)

(9 – x2)

(2x – 6) (y + 3) ÷ (y2 + 2y – 3)

(9 – x2) = (2x – 6) (y + 3)

(y2 + 2y – 3) (9 – x2)*

= 2(x – 3) (y + 3)

(y + 3)(y – 1) (3 – x)(3 + x)*

–1 1

= –2(y – 1) (x + 3)

Rational ExpressionsThe least common denominator (LCD) is needed

Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressions

Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractions

Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractionsIn all these applications, use the LCD as a multiplier to clear denominators.

Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractionsIn all these applications, use the LCD as a multiplier to clear denominators. To Combine Rational Expressions (LCD Method):

Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractionsIn all these applications, use the LCD as a multiplier to clear denominators. To Combine Rational Expressions (LCD Method): To combine rational expressions (F ± G), the numerator of the answer is (LCD)(F ± G) i.e.the answer is (LCD)(F ± G) / LCD.

Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractionsIn all these applications, use the LCD as a multiplier to clear denominators. To Combine Rational Expressions (LCD Method): To combine rational expressions (F ± G), the numerator of the answer is (LCD)(F ± G) i.e.the answer is (LCD)(F ± G) / LCD.

Example H: Combine 712

58 + –

169

Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractionsIn all these applications, use the LCD as a multiplier to clear denominators. To Combine Rational Expressions (LCD Method): To combine rational expressions (F ± G), the numerator of the answer is (LCD)(F ± G) i.e.the answer is (LCD)(F ± G) / LCD.

Example H: Combine 712

58 + –

169

The LCD = 48,

Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractionsIn all these applications, use the LCD as a multiplier to clear denominators. To Combine Rational Expressions (LCD Method): To combine rational expressions (F ± G), the numerator of the answer is (LCD)(F ± G) i.e.the answer is (LCD)(F ± G) / LCD.

Example H: Combine 712

58 + –

169

The LCD = 48, ( ) *48 7

1258

+ – 169

Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractionsIn all these applications, use the LCD as a multiplier to clear denominators. To Combine Rational Expressions (LCD Method): To combine rational expressions (F ± G), the numerator of the answer is (LCD)(F ± G) i.e.the answer is (LCD)(F ± G) / LCD.

Example H: Combine 712

58 + –

169

The LCD = 48, ( ) *48 67

1258

+ – 1694 3

Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractionsIn all these applications, use the LCD as a multiplier to clear denominators. To Combine Rational Expressions (LCD Method): To combine rational expressions (F ± G), the numerator of the answer is (LCD)(F ± G) i.e.the answer is (LCD)(F ± G) / LCD.

Example H: Combine 712

58 + –

169

The LCD = 48, ( ) *48 67

1258

+ – 1694 3

= 28 + 30 – 27 = 31

Rational ExpressionsThe least common denominator (LCD) is needed I. to combine (add or subtract) rational expressionsII. to simplify complex fractionsIn all these applications, use the LCD as a multiplier to clear denominators. To Combine Rational Expressions (LCD Method): To combine rational expressions (F ± G), the numerator of the answer is (LCD)(F ± G) i.e.the answer is (LCD)(F ± G) / LCD.

Example H: Combine 712

58 + –

169

The LCD = 48, ( ) *48 67

1258

+ – 1694 3

= 28 + 30 – 27 = 31

Hence 712

58 + –

169 =

4831

Rational Expressions

– (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3 Example I.

Combine

Rational Expressions

– (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3

y2 + y – 2 = (y – 1)(y + 2)

Example I. Combine

Rational Expressions

– (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3

y2 + y – 2 = (y – 1)(y + 2)

y2 + 2y – 3 = (y – 1)(y + 3)

Example I. Combine

Rational Expressions

– (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3

y2 + y – 2 = (y – 1)(y + 2)

y2 + 2y – 3 = (y – 1)(y + 3)

Hence the LCD = (y – 1)(y + 2)(y + 3),

Example I. Combine

Rational Expressions

– (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3

y2 + y – 2 = (y – 1)(y + 2)

y2 + 2y – 3 = (y – 1)(y + 3)

Hence the LCD = (y – 1)(y + 2)(y + 3), multiply it to the problem:

– (y2 + 2y – 3) (y – 1)(y + 2) 2y – 1 y – 3 [ ] (y – 1)(y + 2)(y + 3)

Example I. Combine

Rational ExpressionsExample I. Combine

– (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3

y2 + y – 2 = (y – 1)(y + 2)

y2 + 2y – 3 = (y – 1)(y + 3)

Hence the LCD = (y – 1)(y + 2)(y + 3), multiply it to the problem:

– (y2 + 2y – 3) (y – 1)(y + 2) 2y – 1 y – 3 [ ] (y – 1)(y + 2)(y + 3)

(y + 3) (y + 2)

Rational Expressions

– (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3

y2 + y – 2 = (y – 1)(y + 2)

y2 + 2y – 3 = (y – 1)(y + 3)

Hence the LCD = (y – 1)(y + 2)(y + 3), multiply it to the problem:

– (y2 + 2y – 3) (y – 1)(y + 2) 2y – 1 y – 3 [ ] (y – 1)(y + 2)(y + 3)

= (2y – 1)(y + 3) – (y – 3)(y + 2)

(y + 3) (y + 2)

Example I. Combine

Rational Expressions

– (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3

y2 + y – 2 = (y – 1)(y + 2)

y2 + 2y – 3 = (y – 1)(y + 3)

Hence the LCD = (y – 1)(y + 2)(y + 3), multiply it to the problem:

– (y2 + 2y – 3) (y – 1)(y + 2) 2y – 1 y – 3 [ ] (y – 1)(y + 2)(y + 3)

= (2y – 1)(y + 3) – (y – 3)(y + 2) = y2 + 6y + 3

(y + 3) (y + 2)

Example I. Combine

Rational Expressions

– (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3

y2 + y – 2 = (y – 1)(y + 2)

y2 + 2y – 3 = (y – 1)(y + 3)

Hence the LCD = (y – 1)(y + 2)(y + 3), multiply it to the problem:

– (y2 + 2y – 3) (y – 1)(y + 2) 2y – 1 y – 3 [ ] (y – 1)(y + 2)(y + 3)

= (2y – 1)(y + 3) – (y – 3)(y + 2) = y2 + 6y + 3

So – (y2 + 2y – 3) (y2 + y – 2) 2y – 1 y – 3 = y2 + 6y + 3

(y – 1)(y + 2)(y + 3)

(y + 3) (y + 2)

Rational ExpressionsA complex fraction is a fraction made with rational expressions.

Rational ExpressionsA complex fraction is a fraction made with rational expressions. To simplify a complex fraction, use thethe LCD to clear all denominators.

Rational Expressions

Example J. Simplify –(x – h)

1

A complex fraction is a fraction made with rational expressions. To simplify a complex fraction, use thethe LCD to clear all denominators.

(x + h)

1

2h

Rational Expressions

Example J. Simplify –(x – h)

1

A complex fraction is a fraction made with rational expressions. To simplify a complex fraction, use thethe LCD to clear all denominators.

(x + h)

1

2h Multiply the top and bottom by (x – h)(x + h) to reduce the expression in the numerators to polynomials.

Rational Expressions

Example J. Simplify –(x – h)

1

A complex fraction is a fraction made with rational expressions. To simplify a complex fraction, use thethe LCD to clear all denominators.

(x + h)

1

2h Multiply the top and bottom by (x – h)(x + h) to reduce the expression in the numerators to polynomials. –(x – h)

1 (x + h)

1

2h = –(x – h)

1 (x + h)

1

2h

(x + h)(x – h)

[

] (x + h)(x –

h)

*

Rational Expressions

Example J. Simplify –(x – h)

1

A complex fraction is a fraction made with rational expressions. To simplify a complex fraction, use thethe LCD to clear all denominators.

(x + h)

1

2h Multiply the top and bottom by (x – h)(x + h) to reduce the expression in the numerators to polynomials. –(x – h)

1 (x + h)

1

2h = –(x – h)

1 (x + h)

1

2h

(x + h)(x – h)

[

] (x + h)(x –

h)

*

= –(x + h)

(x – h) 2h (x + h)(x –

h)

Rational Expressions

Example J. Simplify –(x – h)

1

A complex fraction is a fraction made with rational expressions. To simplify a complex fraction, use thethe LCD to clear all denominators.

(x + h)

1

2h Multiply the top and bottom by (x – h)(x + h) to reduce the expression in the numerators to polynomials. –(x – h)

1 (x + h)

1

2h = –(x – h)

1 (x + h)

1

2h

(x + h)(x – h)

[

] (x + h)(x –

h)

*

= –(x + h)

(x – h) 2h (x + h)(x –

h) = 2h

2h (x + h)(x – h)

Rational Expressions

Example J. Simplify –(x – h)

1

A complex fraction is a fraction made with rational expressions. To simplify a complex fraction, use thethe LCD to clear all denominators.

(x + h)

1

2h Multiply the top and bottom by (x – h)(x + h) to reduce the expression in the numerators to polynomials. –(x – h)

1 (x + h)

1

2h = –(x – h)

1 (x + h)

1

2h

(x + h)(x – h)

[

] (x + h)(x –

h)

*

= –(x + h)

(x – h) 2h (x + h)(x –

h) = 2h

2h (x + h)(x – h)

= 1 (x + h)(x – h)

To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2.

Rationalize Radicals

To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates.

Rationalize Radicals

Example K: Rationalize the numerator

To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates.

hx + h – x

Rationalize Radicals

To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates.

Rationalize Radicals

hx + h – x = h

(x + h – x) (x + h + x) (x + h + x)

*

Example K: Rationalize the numerator hx + h – x

To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates.

Rationalize Radicals

hx + h – x = h

(x + h – x) (x + h + x) (x + h + x)

*

=h

(x + h)2 – (x)2 (x + h + x)

Example K: Rationalize the numerator hx + h – x

To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates.

Rationalize Radicals

hx + h – x = h

(x + h – x) (x + h + x) (x + h + x)

*

=h

(x + h)2 – (x)2 (x + h + x)

Example K: Rationalize the numerator hx + h – x

(x + h) – (x) = h

To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates.

Rationalize Radicals

hx + h – x = h

(x + h – x) (x + h + x) (x + h + x)

*

=h

(x + h)2 – (x)2 (x + h + x)

=h

h(x + h + x)

Example K: Rationalize the numerator hx + h – x

(x + h) – (x) = h

To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates.

Rationalize Radicals

hx + h – x = h

(x + h – x) (x + h + x) (x + h + x)

*

=h

(x + h)2 – (x)2 (x + h + x)

=h

h(x + h + x)

=1

x + h + x

Example K: Rationalize the numerator hx + h – x

(x + h) – (x) = h

Algebraic ExpressionsExercise 1.2 AExpand1. –(2x – 5)(x +3) 2. (3x – 4)(x + 5) – (2x – 5)(x + 3)3. (3x – 4)(2x – 5) – (x + 5)(x + 3)4. (3x – 4)(x + 3) – (x + 5)(2x – 5)Evaluate if x = -3, -2, -1 by using the factored form5. 3x2 – 2x – 1 6. 2x2 – 5x + 27. 4x3 + 3x2 – x 8 . -2x4 + 3x3 + 2x2 Find the signs of the output if x = -3/2, 2/3 9. 3x2 – 2x – 1 10. 2x2 – 5x + 211. 4x3 + 3x2 – x 12 . -2x4 + 3x3 + 2x2

Algebraic Expressions