Teaching Notes 5.14: Factoring the Difference of Squares
Factoring the difference of squares is a skill that requires students to identify the square of a
number and write the square of a variable expression. Both steps can be problem areas.- Explain that a binomial of the forma^2 −b^2 is called the ‘‘difference of squares’’ because it is
the difference of two quantities, each raised to the second power. - Present some examples of square numbers such as 1, 4, 9, 16..., noting that each square
number can be expressed as a number raised to the second power. 1= 12 ,4= 22 ,9= 32 ,
and so on. - Explain that variable expressions raised to an even power can also be expressed as the square
of the base by dividing the exponent by 2. To illustrate this, present examples such asy^4 =
(y^2 )^2 ,x^6 =(x^3 )^2 ,anda^2 b^10 =(ab^5 )^2. - Review the information and example on the worksheet with your students. Make sure that
your students understand how to use the formulaa^2 −b^2 =(a+b)(a−b).
EXTRA HELP:
There is no formula for factoring the sum of two squares.ANSWER KEY:
(1)(x−10)(x+10) (2)(1− 4 x^5 )(1+ 4 x^5 ) (3)(7−y^6 )(7+y^6 ) (4)(3− 10 x)(3+ 10 x)
(5)(8−y)(8+y) (6)(xy−5)(xy+5) (7)(12−x^4 )(12+x^4 ) (8)(x^3 −11)(x^3 +11)
(9)(a^2 b−9)(a^2 b+9) (10)(x^3 y^3 −12)(x^3 y^3 +12) (11)(2x+1)(2x−1) (12)(6x^5 − 5 y)(6x^5 + 5 y)
------------------------------------------------------------------------------------------
(Challenge)Crystal is incorrect. 4+x^16 is the sum of two squares, which cannot be factored
using the formula for factoring the difference of squares.
------------------------------------------------------------------------------------------202 THE ALGEBRA TEACHER’S GUIDE