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Recognizing Structure in Perfect-Square Expressions

9.1.2 Recognizing Structure in Perfect-Square Expressions

9.1.2 • Recognizing Structure in Perfect-Square Expressions

Activity

1. Each expression is written as the product of factors. Write an equivalent expression in standard form.

a. ( 3 x ) 2 ( 3 x ) 2

Compare your answer:

9 x 2 9 x 2

b. 7 x · 7 x 7 x · 7 x

Compare your answer:

49 x 2 49 x 2

c. ( x + 4 ) ( x + 4 ) ( x + 4 ) ( x + 4 )

Compare your answer:

x 2 + 8 x + 16 x 2 + 8 x + 16

d. ( x + 1 ) 2 ( x + 1 ) 2

Compare your answer:

x 2 + 2 x + 1 x 2 + 2 x + 1

e. ( x − 7 ) 2 ( x − 7 ) 2

Compare your answer:

x 2 − 14 x + 49 x 2 − 14 x + 49

f. ( x + n ) 2 ( x + n ) 2

Compare your answer:

x 2 + 2 n x + n 2 x 2 + 2 n x + n 2

2. Why do you think the following expressions can be described as perfect squares?

x 2 + 6 x + 9 x 2 − 16 x + 64 x 2 + 1 3 x + 1 36 x 2 + 6 x + 9 x 2 − 16 x + 64 x 2 + 1 3 x + 1 36

Compare your answer:

If these expressions are written in factored form, each one is the square of a linear expression:

( x + 3 ) 2 ( x + 3 ) 2 , ( x − 8 ) 2 ( x − 8 ) 2 , and ( x + 1 6 ) 2 ( x + 1 6 ) 2 .

Are you ready for more?

Extending Your Thinking
1.

Write each expression in factored form.

x 4 − 30 x 2 + 225 x 4 − 30 x 2 + 225

Compare your answer:

( x 2 − 15 ) 2 ( x 2 − 15 ) 2

2.

x + 14 x + 49 x + 14 x + 49

Compare your answer:

( x + 7 ) 2 ( x + 7 ) 2

3.

5 2 x + 6 · 5 x + 9 5 2 x + 6 · 5 x + 9

Compare your answer:

( 5 x + 3 ) 2 ( 5 x + 3 ) 2

Self Check

Which of the following is equivalent to ( x − 4 ) 2 ?
  1. x 2 + 8 x + 16
  2. x 2 − 8 x + 16
  3. x 2 + 16
  4. x 2 − 16

Additional Resources

Creating Perfect Square Trinomials

A perfect square trinomial is created by multiplying a binomial by itself.

Example

Write ( x − 2 ) 2 ( x − 2 ) 2 as an expanded polynomial in standard form:

Step 1 - Multiply the binomial by itself.

( x − 2 ) ( x − 2 ) ( x − 2 ) ( x − 2 )

Step 2 - Use the distributive property to expand (FOIL).

x 2 − 2 x − 2 x + 4 x 2 − 2 x − 2 x + 4

Step 3 - Combine like terms.

x 2 − 4 x + 4 x 2 − 4 x + 4

Try it

Creating Perfect Square Trinomials

Write ( x + 5 ) 2 ( x + 5 ) 2 as an expanded polynomial in standard form.

Here is how to expand the polynomial:

Step 1 - Multiply the binomial by itself.
( x + 5 ) ( x + 5 ) ( x + 5 ) ( x + 5 )

Step 2 - Use the distributive property (FOIL).
x 2 + 5 x + 5 x + 25 x 2 + 5 x + 5 x + 25

Step 3 - Combine like terms.
x 2 + 10 x + 25 x 2 + 10 x + 25