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Unit 10.2 Factoring Trinomials and Special Cases

Difference of Squares

The difference of squares is a specific algebraic identity that states that the difference between the squares of two numbers can be factored into a product of two binomials. It is expressed as

Difference of Squares

a2 – b2 = (a + b)(a – b)

For the difference of squares, it must be a minus sign and should not be confused with (a2 + b2) which cannot be factored in this method

Factoring using difference of squares

factor x2 – 9

9 is a perfect square of 3 so we can rewrite it as

x2 – 32

which matches the a2 – bformat. We are able to factor it

(x – 3)(x + 3)

 

Sum and Difference of Cubes

The sum and difference of cubes are specific algebraic identities that help in factoring expressions of the form (a3 + b3) or (a3 – b3)

Sum of Cubes

a3 + b3 = (a + b) (a2 – ab + b2)

Difference of Cubes

a3 – b3 = (a – b) (a2 + ab + b2)

Factoring using difference of cubes

Factor 27y3 – 1

27 , y3 and 1 are all perfect cubes so we are able to rewrite it as

(3y)3 – (1)3

Which matches the a3 – b3 format. We are able to factor it using a3 – b3 = (a – b) (a2 + ab + b2)

(3y)3 – (1)= (3y – 1) ((3y)2 + (3y)(1) + 12)

= (3y – 1) (9y2 + 3y + 1)

 

 

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