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Factoring Differences of Squares. Multiplying Conjugates. The following pairs of binomials are called conjugates . Notice that they all have the same terms, only the sign between them is different. (3x + 6). and. (3x - 6). (r - 5). and. (r + 5). (2b - 1). and. (2b + 1). (x 2 + 5).

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Factoring

Differences

of Squares


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Multiplying Conjugates

The following pairs of binomials are called conjugates. Notice that they all have the same terms, only the sign between them is different.

(3x + 6)

and

(3x - 6)

(r - 5)

and

(r + 5)

(2b - 1)

and

(2b + 1)

(x2 + 5)

and

(x2 - 5)


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x2

x

x

x

x

-x

-x

-x

-x

Multiplying Conjugates

Multiply: (x + 4)(x – 4) using algebra tiles.

x + 4

x

-

4

= x • x + x • (-4) + 4 • x + 4 • (-4)

FOIL: (x + 4)(x – 4)

= x2 + (-4x) + 4x + (-16)


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x + 4

x2

x

-

4

x

x

x

x

-x

-x

-x

-x

Multiplying Conjugates

Multiply: (x + 4)(x – 4) using algebra tiles.

Opposite tiles add up to zero (or cancel). Cancel out any opposite pairs!

= x • x + x • (-4) + 4 • x + 4 • (-4)

FOIL: (x + 4)(x – 4)

= x2 + (-4x) + 4x + (-16)


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x2

x

x

x

-x

-x

-x

Multiplying Conjugates

Multiply: (x + 4)(x – 4) using algebra tiles.

x + 4

x

-

4

Opposite tiles add up to zero (or cancel). Cancel out any opposite pairs!

= x • x + x • (-4) + 4 • x + 4 • (-4)

FOIL: (x + 4)(x – 4)

= x2 + (-4x) + 4x + (-16)


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x2

x

x

-x

-x

Multiplying Conjugates

Multiply: (x + 4)(x – 4) using algebra tiles.

x + 4

x

-

4

Opposite tiles add up to zero (or cancel). Cancel out any opposite pairs!

= x • x + x • (-4) + 4 • x + 4 • (-4)

FOIL: (x + 4)(x – 4)

= x2 + (-4x) + 4x + (-16)


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x2

x

-x

Multiplying Conjugates

Multiply: (x + 4)(x – 4) using algebra tiles.

x + 4

x

-

4

Opposite tiles add up to zero (or cancel). Cancel out any opposite pairs!

= x • x + x • (-4) + 4 • x + 4 • (-4)

FOIL: (x + 4)(x – 4)

= x2 + (-4x) + 4x + (-16)


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x2

Multiplying Conjugates

Multiply: (x + 4)(x – 4) using algebra tiles.

x + 4

x

-

4

Opposite tiles add up to zero (or cancel). Cancel out any opposite pairs!

= x • x + x • (-4) + 4 • x + 4 • (-4)

FOIL: (x + 4)(x – 4)

= x2 + (-4x) + 4x + (-16)


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x2

Multiplying Conjugates

Multiply: (x + 4)(x – 4) using algebra tiles.

x + 4

x

-

4

= x2 + (-16)

= x • x + x • (-4) + 4 • x + 4 • (-4)

FOIL: (x + 4)(x – 4)

= x2 + (-4x) + 4x + (-16)


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x2

Multiplying Conjugates

Multiply: (x + 4)(x – 4) using algebra tiles.

x + 4

x

-

4

Opposite terms also add up to zero (or cancel). Cancel out any opposite pairs!

= x • x + x • (-4) + 4 • x + 4 • (-4)

FOIL: (x + 4)(x – 4)

= x2 + (-4x) + 4x + (-16)


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x2

Multiplying Conjugates

Multiply: (x + 4)(x – 4) using algebra tiles.

x + 4

x

-

4

Opposite terms also add up to zero (or cancel). Cancel out any opposite pairs!

= x • x + x • (-4) + 4 • x + 4 • (-4)

FOIL: (x + 4)(x – 4)

= x2 + (-4x) + 4x + (-16)

= x2 + 0 + (-16)

= x2 + (-16)


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Multiplying Conjugates

When we multiply any conjugate pairs, the middle terms always cancel and we end up with a binomial.

(3x + 6)(3x - 6)

= 9x2 - 36

(r - 5)(r + 5)

= r2 - 25

= 4b2 - 1

(2b - 1)(2b + 1)


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A MINUS between!

Difference of Squares

Binomials that look like this are called a Difference of Squares:

Only TWO terms (a binomial)

9x2 - 36

The first term is a Perfect Square!

The second term is a Perfect Square!


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A Difference of Squares!

A Conjugate Pair!

Factor a Difference of Squares:


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Factor a Difference of Squares:

= (x + 8)(x - 8)

Example: Factor x2 - 64

x2 = x • x

64 = 8 • 8

= (3t + 5)(3t - 5)

Example: Factor 9t2 - 25

9t2 = 3t • 3t

25 = 5 • 5


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A Sum of Squares?

A Sum of Squares, like x2 + 64, can NOT be factored!

It is a PRIME polynomial.


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Practice

Factor each polynomial.

1) x2 - 81

2) r2 - 100

3) 16 - a2

4) 9a2 - 16

5) 16x2 - 1


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Practice - Answers

Factor each polynomial.

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

2) r2 - 100 = (r + 10)(r - 10)

3) 16 - a2 = (4 + a)(4 - a)

4) 9a2 - 16 = (3a + 4)(3a - 4)

5) 16x2 - 1 = (4x + 1)(4x - 1)