Algebra 1 Factoring special products

CONFIDENTIAL 1

Algebra 1Algebra 1

Factoring special Factoring special productsproducts

CONFIDENTIAL 2

Warm UpWarm Up

1) 10x2 + 11x + 3

Factor the trinomial. Check your answer.

2) 9x2 + 17x + 8

3) 5x2 + 19x + 18

4) 4x2 - 12x + 9

1) (x + 5)(x + 6)

2) (x + 8)(x + 9)

3) (x + 1)(x + 18)

4) (2x - 3)(2x - 3)

CONFIDENTIAL 3

To factor ax2 + bx + c, check the factors of a and the factors of c in the binomials. the sum of the products

of the outer and inner terms should be b.

Since you need to check all the factors of a and all the factors of c, it may be helpful to make the table.

Then check the products of outer and the inner terms to see if the sum is b. You can multiply the

binomials to check your answer.

product =a

product =c

Sum of inner and outer products = b

( x + )( x + ) = ax2 + bx + c

Let’s review what we did in the last session

CONFIDENTIAL 4

Factoring axx22 + bx + c + bx + c when c is positiveFactor the trinomial. Check your answer.

2x2 + 11x + 12

Factors of 2 Factors of 12 outer + inner

1 and 21 and 21 and 21 and 21 and 21 and 2

(x + 4)(2x + 3).

Check: (x + 4)(2x + 3) = 2x2 + 3x + 8x + 12 = 2x2 + 11x + 12

( x + )( x + ) a = 2 and c = 12; outer + inner = 11.


1(12) + 2(1) = 141(1) + 2(12) = 251(6) + 2(2) = 101(2) + 2(6) = 141(4) + 2(3) = 101(3) + 2(4) = 11

Use the FOIL method.

CONFIDENTIAL 5

Factoring axx22 + bx + c + bx + c when c is negativeFactor the trinomial. Check your answer.

2x2 - 7x - 15

Check: (x - 5)(2x + 3) = 2x2 + 3x - 10x - 15 = 2x2 - 7x - 15

Use the FOIL method.

( x + )( x + ) a = 2 and c = -15; outer + inner = -7.

(x - 5)(2x + 3).

Factors of 2 Factors of -15 outer + inner


1 and -15-1 and 153 and -5-3 and 55 and -3 -5 and 3

1(-15) + 1(2) = -131(15) + 2(-1) = 131(-5) + 2(3) = 11(5) + 2(-3) = -11(-3) + 2(5) = 71(3) + 2(-5) = -7

CONFIDENTIAL 6

When the leading coefficient is negative, factor out -1 from each term before using factoring methods.

Factor -2x2 - 15x - 7.

-1(2x2 + 15x + 7)

Factoring axx22 + bx + c + bx + c when a is negative

( x + )( x + )

Factor out -1

a = 2 and c = 7; outer + inner = 15.

(x + 7)(2x + 1).


1 and 21 and 2

1 and 77 and 1

(1)7 + 2(1) = 91(1) + 2(7) = 15

-1(x + 7)(2x + 1).

Check: -1(x + 7)(2x + 1)= -1(2x2 + x + 14x + 7) = -2x2 - 15x - 7

CONFIDENTIAL 7

The area of a rectangle in square feet can be represented by 6x2 + 11x + 5. The width is (x + 1) ft. What is the

length of the rectangle?

6x2 + 11x + 5

( x + )( x + )

a = 6 and c = 5; outer + inner = 11.


1 and 62 and 33 and 26 and 1

1 and 55 and 11 and 55 and 1

1(5) + 6(1) = -132(1) + 3(5) = 133(5) + 2(1) = 16(1) + 1(5) = -1

6x2 + 11x + 5 = (6x + 5)(x + 1).

Width of the rectangle = (6x + 5)

CONFIDENTIAL 8

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 start

CONFIDENTIAL 9

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 10

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 11

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 12

Now you try!

1) 4x2 - 12x + 9

2) 9x2 - 6x + 4


1) Yes. (x - 3)2

2) No

CONFIDENTIAL 13

Problem solving application

Many Texas courthouses are at the center of a town square. The area of the town square is

(25 x 2 + 70x + 49) ft2 .

The dimensions of the square are approximately cx + d, where c and d are whole numbers.

a) Write an expression for the perimeter of the town square.

b) Find the perimeter when x = 60.

The town square is a rectangle with area = (25x2 + 70x + 49) ft2 .

The dimensions of the town square are of the form (cx + d) ft2 , where c and d are whole numbers.

SOLUTION:

Next page

CONFIDENTIAL 14

The formula for the area of a rectangle is area = length × width.

Factor (25x2 + 70x + 49) to find the length and width of the town square.

Write a formula for the perimeter of the town square, and evaluate the expression for x = 60.

25x2 + 70x + 49

= (5x)2 + 2(5x)(7) + 72

= (5x + 7)2

a = 5x, b = 7


Write the trinomial as (a + b)2

25x2 + 70x + 49 = (5x + 7)(5x + 7)

The length and width of the town square are (5x + 7) ft and (5x + 7) ft.

Next page

CONFIDENTIAL 15

Because the length and width are equal, the town square is a square.

The perimeter of the town square = 4s= 4 (5x + 7)= 20x + 28

Substitute the side length for s.

Distribute 4.

a) An expression for the perimeter of the town square in feet is (20x + 28).

Evaluate the expression when x = 60.

P = 20x + 28 = 20 (60) + 28 = 1228

Substitute 60 for x.

b) When x = 60, the perimeter of the town square is 1288 ft.

CONFIDENTIAL 16

Now you try!

A company produces square sheets of aluminum, each of which has an area of (9x2 + 6x + 1) m2 . The

side length of each sheet is in the form cx + d, where c and d are whole numbers.

a) Find an expression in terms of x for the perimeter

of a sheet.

b) Find the perimeter when x = 3 m.

a) An expression for the perimeter of the sheet (12x + 4).

b) When x = 3, the perimeter of the sheet is 40 ft.

CONFIDENTIAL 17

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 18

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 19

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 20

Now you try!

1) 9p4 - 16q2

2) 16x2 - 4y5

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

1) Yes. (3p2 + 4q) (3p2 - 4q)

2) No. 4y5 is not a perfect square.

CONFIDENTIAL 21

Assessment

1) x2 - 4x + 4

2) x2 - 4x - 4

3) 9x2 - 12x + 4

4) x2 + 2x + 1


5) x2 - 6x + 6

1) Yes. (x - 2)2

2) No

3) Yes. (3x - 2)2

4) Yes. (x + 1)2

5) No

CONFIDENTIAL 22

6) 1 - 4x2

7) p7 - 49q6

8) 4x2 - 12

9) 16x2 - 225

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

6) (1 + 2x)(1 - 2x)

7) No

8) No

9) (4x + 15)(4x - 15)

CONFIDENTIAL 23

10) A city purchases a rectangular plot of land with an area of ( x2 + 24x + 144) yd2 for a park. The dimensions

of the plot are of the form ax + b, where a and b are whole numbers.

a) Find an expression for the perimeter of the park.

b) Find the perimeter when x = 10 yd.

a) An expression for the perimeter of the park (4x + 48).

b) When x = 10, the perimeter of the park is 88 ft.

CONFIDENTIAL 24

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

CONFIDENTIAL 25

Recognizing and factoring perfect perfect square square trinomials


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


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

Next page

CONFIDENTIAL 26

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 27

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 28

Problem solving application

Many Texas courthouses are at the center of a town square. The area of the town square is

(25 x 2 + 70x + 49) ft2 .

The dimensions of the square are approximately cx + d, where c and d are whole numbers.

a) Write an expression for the perimeter of the town square.

b) Find the perimeter when x = 60.

The town square is a rectangle with area = (25x2 + 70x + 49) ft2 .

The dimensions of the town square are of the form (cx + d) ft2 , where c and d are whole numbers.

SOLUTION:

Next page

CONFIDENTIAL 29

The formula for the area of a rectangle is area = length × width.

Factor (25x2 + 70x + 49) to find the length and width of the town square.

Write a formula for the perimeter of the town square, and evaluate the expression for x = 60.

25x2 + 70x + 49

= (5x)2 + 2(5x)(7) + 72

= (5x + 7)2

a = 5x, b = 7


Write the trinomial as (a + b)2

25x2 + 70x + 49 = (5x + 7)(5x + 7)

The length and width of the town square are (5x + 7) ft and (5x + 7) ft.

Next page

CONFIDENTIAL 30

Because the length and width are equal, the town square is a square.

The perimeter of the town square = 4s= 4 (5x + 7)= 20x + 28

Substitute the side length for s.

Distribute 4.

a) An expression for the perimeter of the town square in feet is (20x + 28).

Evaluate the expression when x = 60.

P = 20x + 28 = 20 (60) + 28 = 1228

Substitute 60 for x.

b) When x = 60, the perimeter of the town square is 1288 ft.

CONFIDENTIAL 31

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 32

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 33

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 34

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Algebra 1 Factoring special products

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