Factoring Based on Conjugates

Factoring Based on Conjugates

Algebra 2

There are a number of different types of factoring techniques.

• Each one of them boils down to reversing a product

• Conjugate binomials, or binomials of the form  $a+b$  and  $a-b$  

Multiple Choice

Find the product

 $\left(x+5\right)\left(x-5\right)$  

•  $x^2-25$  

•  $x^2+10x+25$  

•  $x^2-4$  

•  $x^2+4x+4$  

Multiple Choice

Find the product

 $\left(4x-1\right)\left(4x+1\right)$  

•  $16x^2+16x+1$  

•  $16x^2-16x+1$  

•  $16x^2-1$  

•  $16x^2+1$  

Fill in the Blanks

Find the product

 $\left(x-y\right)\left(x+y\right)$ 

Do not include any spaces in your answer 

Multiple Choice

Find the product

 $\left(7x+2y\right)\left(7x-2y\right)$  

•  $49x^2+4y^2$  

•  $49x^2+28xy+4y^2$  

•  $49x^2-28xy+4y^2$  

•  $49x^2-4y^2$  

What we should see is that if we multiply conjugates, opposites always cancel and instead of getting our expected trinomial, we still get a binomial.

 $\left(a-b\right)\left(a+b\right)=a^2-b^2$  

We can use this pattern to quickly find products that match this format

Fill in the Blanks

Find the product

 $\left(x-6\right)\left(x+6\right)$  

Remember:  $\left(a-b\right)\left(a+b\right)=a^2-b^2$  
Do not include spaces in your answer

Fill in the Blanks

Find the product

 $\left(5x-2\right)\left(5x+2\right)$  

Remember:  $\left(a-b\right)\left(a+b\right)=a^2-b^2$  
Do not include spaces in your answer

Poll

Does it matter if addition or subtraction comes first?

Ex:  $\left(a+b\right)\left(a-b\right)$  is not the same as  $\left(a-b\right)\left(a+b\right)$  

• Yes it matters

• No it does not matter

Fill in the Blanks

Find the product

 $\left(4+x\right)\left(4-x\right)$  

Remember:  $\left(a-b\right)\left(a+b\right)=a^2-b^2$  
Do not include spaces in your answer

Fill in the Blanks

Find the product

 $\left(6-5x\right)\left(6+5x\right)$  

Remember:  $\left(a-b\right)\left(a+b\right)=a^2-b^2$  
Do not include spaces in your answer

We now should be able to reverse this multiplication in order to rewrite expressions that are the difference of perfect squares into products.

Multiple Choice

Find the correct factored form of

 $x^2-81$  

•  $\left(x-7\right)\left(x+7\right)$  

•  $\left(x-9\right)\left(x+9\right)$  

•  $\left(x+81\right)^2$  

•  $\left(9-x\right)\left(9+x\right)$  

Multiple Choice

Find the correct factored form of

 $9x^2-4$  

•  $\left(3x-4\right)\left(3x+4\right)$  

•  $\left(6x-2\right)\left(6x+2\right)$  

•  $\left(3x-2\right)^2$  

•  $\left(3x-2\right)\left(3x+2\right)$  

Fill in the Blanks

Find the correct factored form of

 $25-x^2$  

Do not include any spaces in your answer

Fill in the Blanks

Find the correct factored form of

 $100x^2-1$  

Do not include any spaces in your answer