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


Expressions

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


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.


Expressions

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



Expressions

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


Algebraic Expressions

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




Examples of algebraic expressions are

3x2 – 2x + 4,




3x2 – 2x + 4,x2 + 3

3 x3 – 2x – 4 ,




3x2 – 2x + 4,x2 + 3

3 x3 – 2x – 4 ,

(x1/2 + y)1/3

(4y2 – (x + 4)1/2)1/4




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).




3x2 – 2x + 4,x2 + 3

3 x3 – 2x – 4 ,

(x1/2 + y)1/3

(4y2 – (x + 4)1/2)1/4


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




3x2 – 2x + 4,x2 + 3

3 x3 – 2x – 4 ,

(x1/2 + y)1/3

(4y2 – (x + 4)1/2)1/4


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.


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


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


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

Insert [ ]


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 [ ]


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 [ ]



Insert [ ]

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



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


= (4x + 5)((4x)2 – (4x)(5) +(5)2)



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



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


We factor polynomials for the following purposes.


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



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


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



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.


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



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.


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)


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]


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


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]


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


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]


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


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.


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


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)


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 E. Factor

x2 – 1 x2 – 3x+ 2





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.


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


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.


Example F. Reduce x2 – 1 x2 – 3x+ 2

x*yx*z = x*y

x*z = yz



i.e.



x2 – 1 x2 – 3x+ 2 = (x – 1)(x + 1)

(x – 1)(x – 2)

x*yx*z = x*y

x*z = yz


factor


i.e.



x2 – 1 x2 – 3x+ 2 = (x – 1)(x + 1)

(x – 1)(x – 2)

x*yx*z = x*y

x*z = yz


= (x + 1) (x – 2)

factor


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



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)



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)*



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)*



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



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.


Example H: Combine 712

58 + –

169



58 + –

169

The LCD = 48,



58 + –

169

The LCD = 48, ( ) *48 7

1258

+ – 169



58 + –

169

The LCD = 48, ( ) *48 67

1258

+ – 1694 3



58 + –

169

The LCD = 48, ( ) *48 67

1258

+ – 1694 3

= 28 + 30 – 27 = 31



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


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

y2 + y – 2 = (y – 1)(y + 2)

Example 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)

Example 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),

Example 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)

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)


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

(y + 3) (y + 2)


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

y2 + y – 2 = (y – 1)(y + 2)

y2 + 2y – 3 = (y – 1)(y + 3)


– (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


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

y2 + y – 2 = (y – 1)(y + 2)

y2 + 2y – 3 = (y – 1)(y + 3)


– (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


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

y2 + y – 2 = (y – 1)(y + 2)

y2 + 2y – 3 = (y – 1)(y + 3)


– (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.


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



1


(x + h)

1

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



1


(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)

*



1


(x + h)

1


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)



1


(x + h)

1


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)

1


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.


Example K: Rationalize the numerator


hx + h – x




hx + h – x = h

(x + h – x) (x + h + x) (x + h + x)

*

Example K: Rationalize the numerator hx + h – x



hx + h – x = h

(x + h – x) (x + h + x) (x + h + x)

*

=h

(x + h)2 – (x)2 (x + h + x)




hx + h – x = h

(x + h – x) (x + h + x) (x + h + x)

*

=h

(x + h)2 – (x)2 (x + h + x)


(x + h) – (x) = h



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)


(x + h) – (x) = h



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


(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

1.2 algebraic expressions

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