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Polynomial Multiplication:

Product of Conjugates

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What are Conjugates?

ba

ba

The first binomial is the sum of a and b. The conjugate binomial is the difference of a and b.

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Let’s Look at some Examples!

Binomial Conjugate

5x 5x8x 8x

12 x 12 x3 x 3 x73 x 73 x

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Expression Conjugate

Will there be more conjugates in Algebra

?

Yes. Have a look at a couple examples that come up in more advanced sections:

273 x 273 x

i54 i54

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The Product of Conjugates

Use FOIL to multiply these conjugates.

)ax)(ax( 2x ax ax 2a

22 ax

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FOIL Conjugatesusing a Generic

Rectangle

)ax)(ax(

x a

xa

2xax

ax2a

2x ax ax 2a22 ax

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The Shortcut!

)ax)(ax(

22 ax

2x ax ax 2a

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The Shortcut:Example 1

)x)(x( 33 22 3x

92 x

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The Shortcut:Example 2

)x)(x( 77 22 7x

492 x

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The Shortcut:Example 3

)x)(x( 5252

22 52 x

254 2 x

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The Shortcut:Example 4

)x)(x( 6767

22 67 x

3649 2 x

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The Shortcut:Example 5

)x)(x( 112112

22 112 x

1144 2 x

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The Shortcut:Example 6

)yx)(yx( 9494

22 94 yx 22 8116 yx

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Memorize this Pattern

for now and for later!

)ax)(ax( 22 ax In a future section, we will need to work this problem

in reverse. We will be given the difference of

squares and must rewrite it as a product of

conjugates. It will look like this: )x)(x( 33 92x

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That’s All for Now!