confidential 1 algebra i choosing a factoring method

CONFIDENTIAL 1

Algebra IAlgebra I

Choosing a FactoringChoosing a FactoringMethodMethod

CONFIDENTIAL 2

Warm UpWarm Up

1) x2 - 4x + 4

2) x2 - 4x - 4

3) 4x2 - 12

4) 16x2 - 225

Determining whether the trinomial/binomial is a perfect square. If so, factor. If not, explain:

CONFIDENTIAL 3

A trinomial is a perfect square if: The first and the last terms are perfect squares. The middle term is two times one factor from the

first term and one factor from the last term.

9x2 + 12x + 4

3x.3x 2.22(3x.2)

Perfect square trinomials Examples

a2 + 2ab + b2 = (a + b) (a + b) = (a + b)2

x2 + 6x + 9 = (x + 3) (x + 3) = (x + 3)2

a2 - 2ab + b2 = (a - b) (a - b) = (a - b)2

x2 - 6x + 9 = (x - 3) (x - 3) = (x - 3)2

Let’s review what we did in the last session

CONFIDENTIAL 4

Recognizing and factoring perfect perfect square square trinomials

Determining whether the trinomial is a perfect square. If so, factor. If not, explain:

1) x2 + 12x + 36

x2 + 12x + 36

x.x 6.62(x . 6)The trinomial is a perfect square.

METHOD 1: Use the rule.

x2 + 12x + 36

= x2 + 2(x.6) + (6)2

a = x; b = 6

Write the trinomial as a2 + 2ab + b2

= (x + 6)2Write the trinomial as (a + b)2

Next page

CONFIDENTIAL 5

METHOD 2: Factor.

x2 + 12x + 36

Factors of 36 sum

1 and 362 and 183 and 124 and 96 and 6

2514111012

= (x + 6)2

(x + 6)(x + 6)

CONFIDENTIAL 6

2) x2 + 9x + 16

x2 + 9x + 16

x.x 4.42(x . 4) 2(x . 4) = 9x

x2 + 9x + 16 is not a perfect square because 2(x . 4) = 9x.

CONFIDENTIAL 7

The difference of two squares (a2 - b2) can be written as the product (a + b) (a - b) .

You can use this pattern to factor some polynomials.

A polynomial is a difference of two squares if:

There are two terms, one subtracted from the other. Both terms are perfect squares.

4x2 - 9

2x · 2x 3 · 3

DIFFERENCE OF TWO SQUARES EXAMPLE

a2 - b2 = (a + b) (a - b) x2 - 9 = (x + 3) (x - 3)

(a2 - b2)

CONFIDENTIAL 8

Recognizing and Factoring the Difference of Two Squares

Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.

1) x6 - 7y2

x2 - 81

x · x 9 · 9

The polynomial is a difference of two squares.

x2 - 92 a = x, b = 9

= (x + 9)(x - 9) Write the polynomial as (a + b) (a - b) .

x2 - 81 = (x + 9) (x - 9)

CONFIDENTIAL 9

2) x2 - 7y2

x6 - 7y2

x3 · x3 7y2 is not a perfect square.

x6 - 7y2 is not the difference of two squares because 7y2 is not a perfect square.

CONFIDENTIAL 10

Let’s start

Solving an equation that involves that polynomial may require factoring the polynomial. A polynomial

is in its fully factored form when it is written as aproduct that cannot be factored further.

Determining Whether a Polynomial Is Completely Factored

Tell whether the polynomial (2x + 6) (x + 5) is completely factored. If not, factor it.

(2x + 6) (x + 5)

=2 (x + 3) (x + 5)

2x + 6 can be further factored.

2 (x + 3) (x + 5) is completely factored.

Factor out 2, the GCF of 2x and 6.

CONFIDENTIAL 11

To factor a polynomial completely, you may need to use more than one factoring method. Use the steps

below to factor a polynomial completely.

Factoring Polynomials

Step 1: Check for a greatest common factor.

Step 2: Check for a pattern that fits the difference of two squares or a perfect-square trinomial.

Step 3: To factor x2 + bx + c, look for two numbers whose sum is b and whose product is c. To factor a x2 + bx + c, check factors of a and factors of c in the binomial factors. The sum of the products of the outer and inner terms should be b.

Step 4: Check for common factors.

CONFIDENTIAL 12

Now you try!

1) 5x2(x - 1)

2) (4x + 4) (x + 1)

Tell whether the polynomial is completely factored. If not, factor it.

CONFIDENTIAL 13

Factoring by GCF and Recognizing Patterns

Factor -2xy2 + 16xy - 32x completely. Check your answer.

-2xy2 + 16xy - 32x

= -2x(y2 - 8y + 16)

=-2x(y - 4)2

Factor out the GCF. y2 - 8y + 16 is a perfect square trinomial

of the form a2 - 2ab + b2.

a = y, b = 4

Check:

-2x(y - 4)2 = -2x(y2 - 8y + 16)

-2xy2 + 16xy - 32x

If none of the factoring methods work, the polynomial is said to be unfactorable.

CONFIDENTIAL 14

Factor each polynomial completely. Check your answer.

Now you try!

1) 4x3 + 16x2 + 16x

2) 2x2y - 2y3

CONFIDENTIAL 15

Factoring by Multiple Methods

Factor each polynomial completely.

1) 2x2 + 5x + 4

( x + )( x + ) The GCF is 1 and there is no pattern.

a = 2 and c = 4; outer + inner = 5.

Factors of 2 Factors of 4 outer + inner

1 and 21 and 21 and 2


1(4) + 2(1) = 61(1) + 2(4) = 91(2) + 2(2) = 6

2x2 + 5x + 4 is unfactorable.

CONFIDENTIAL 16

2) 3n4 - 15n3 + 12n2

3n2(n2 - 5n + 4)

(x + )( x + ) Factor out the GCF. There is no pattern.

b = -5 and c = 4; look for factors of 4 whose sum is -5.

Factors of 4 Sum

-1 and -4-2 and -2

-5-4

The factors needed are -1 and -4.

3n4 - 15n3 + 12n2 = 3n2(n - 1)(n - 4)

CONFIDENTIAL 17

3) p5 - p

p(p4 - 1)

=p(p2 + 1)(p2 - 1)

=p(p2 + 1)(p + 1)(p - 1)

Factor out the GCF.

p4 - 1 is a difference of two squares.


CONFIDENTIAL 18


Now you try!

1) 3x2 + 7x + 4

2) 2p5 + 10p4 - 12p3

3) 9q6 + 30q5 + 24q4

CONFIDENTIAL 19

Methods to Factor Polynomials

Any Polynomial—Look for the greatest common factor.

ab - ac = a(b - c) 6x2y + 10xy2 = 2xy (3x + 5y)

Binomials—Look for a difference of two squares.

a2 - b2 = (a + b)(a - b) x2 - 9y2 = (x + 3y)(x - 3y)

Trinomials—Look for perfect-square trinomials and other factorable trinomials.

a2 + 2ab + b2 = (a + b)2

a2 - 2ab + b2 = (a - b)2

x2 + 4x + 4 = (x + 2)2

x2 - 4x + 4 = (x - 2)2

CONFIDENTIAL 20


x2 + bx + c = (x + )(x + )x2 + bx + c = ( x + )( x + )

x2 + 3x + 2 = (x + 1)(x + 2)6x2 + 7x + 2= (2x+1)

(3x+2)

Polynomials of Four or More Terms—Factor by grouping.

ax + bx + ay + by = x(a + b) + y(a + b)

= (x + y) (a + b)

2x3+4x2 +x+ 2=(2x3 + 4x2) + (x + 2)

= 2x2(x + 2) + 1(x + 2)= (x + 2)(2x2 + 1)

CONFIDENTIAL 21

BREAK

CONFIDENTIAL 22Title: Format: WebEx Web BrowserDouble-click to edit

CONFIDENTIAL 23

Assignments

1) 2x (x2 + 4)

2) 3x(9x2 - 1)

3) 4x3 - 4x2 - 8x

Tell whether each polynomial is completely factored. If not, factor it:

CONFIDENTIAL 24

4) 4x3 + 18x2 + 20x

5) 2x4 + 18

6) 3x5 - 12x3

7) 4x3 + 8x2 + 4x


CONFIDENTIAL 25

8) The square of Ella’s age plus 12 times Ella’s age plus 36.

Write an expression for each situation. Factor your expression.

9) The square of the distance from point A to point B minus 81.

10) Factor and simplify: (2x + 3)2 - (x - 4)2

CONFIDENTIAL 26

Solving an equation that involves that polynomial may require factoring the polynomial. A polynomial

is in its fully factored form when it is written as aproduct that cannot be factored further.

Determining Whether a Polynomial Is Completely Factored

Tell whether the polynomial (2x + 6) (x + 5) is completely factored. If not, factor it.

(2x + 6) (x + 5)

=2 (x + 3) (x + 5)

2x + 6 can be further factored.

2 (x + 3) (x + 5) is completely factored.

Factor out 2, the GCF of 2x and 6.

Let’s review

CONFIDENTIAL 27

To factor a polynomial completely, you may need to use more than one factoring method. Use the steps

below to factor a polynomial completely.

Factoring Polynomials

Step 1: Check for a greatest common factor.

Step 2: Check for a pattern that fits the difference of two squares or a perfect-square trinomial.

Step 3: To factor x2 + bx + c, look for two numbers whose sum is b and whose product is c. To factor a x2 + bx + c, check factors of a and factors of c in the binomial factors. The sum of the products of the outer and inner terms should be b.

Step 4: Check for common factors.

CONFIDENTIAL 28

Factoring by GCF and Recognizing Patterns

Factor -2xy2 + 16xy - 32x completely. Check your answer.

-2xy2 + 16xy - 32x

= -2x(y2 - 8y + 16)

=-2x(y - 4)2

Factor out the GCF. y2 - 8y + 16 is a perfect square trinomial

of the form a2 - 2ab + b2.

a = y, b = 4

Check:

-2x(y - 4)2 = -2x(y2 - 8y + 16)

-2xy2 + 16xy - 32x

If none of the factoring methods work, the polynomial is said to be unfactorable.

CONFIDENTIAL 29

Factoring by Multiple Methods

Factor each polynomial completely.

1) 2x2 + 5x + 4

( x + )( x + ) The GCF is 1 and there is no pattern.

a = 2 and c = 4; outer + inner = 5.

Factors of 2 Factors of 4 outer + inner



1(4) + 2(1) = 61(1) + 2(4) = 91(2) + 2(2) = 6

2x2 + 5x + 4 is unfactorable.

CONFIDENTIAL 30

2) 3n4 - 15n3 + 12n2

3n2(n2 - 5n + 4)

(x + )( x + ) Factor out the GCF. There is no pattern.

b = -5 and c = 4; look for factors of 4 whose sum is -5.

Factors of 4 Sum

-1 and -4-2 and -2

-5-4

The factors needed are -1 and -4.

3n4 - 15n3 + 12n2 = 3n2(n - 1)(n - 4)

CONFIDENTIAL 31

3) p5 - p

p(p4 - 1)

=p(p2 + 1)(p2 - 1)

=p(p2 + 1)(p + 1)(p - 1)

Factor out the GCF.



CONFIDENTIAL 32

Methods to Factor Polynomials

Any Polynomial—Look for the greatest common factor.

ab - ac = a(b - c) 6x2y + 10xy2 = 2xy (3x + 5y)

Binomials—Look for a difference of two squares.

a2 - b2 = (a + b)(a - b) x2 - 9y2 = (x + 3y)(x - 3y)


a2 + 2ab + b2 = (a + b)2

a2 - 2ab + b2 = (a - b)2

x2 + 4x + 4 = (x + 2)2

x2 - 4x + 4 = (x - 2)2

CONFIDENTIAL 33


x2 + bx + c = (x + )(x + )x2 + bx + c = ( x + )( x + )

x2 + 3x + 2 = (x + 1)(x + 2)6x2 + 7x + 2= (2x+1)

(3x+2)

Polynomials of Four or More Terms—Factor by grouping.

ax + bx + ay + by = x(a + b) + y(a + b)

= (x + y) (a + b)

2x3+4x2 +x+ 2=(2x3 + 4x2) + (x + 2)

= 2x2(x + 2) + 1(x + 2)= (x + 2)(2x2 + 1)

CONFIDENTIAL 34

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confidential 1 algebra i choosing a factoring method

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