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Chapter 6: Problem 217

Factor completely. $$ 16 x^{2}-72 x+81 $$

Short Answer

Expert verified
The factored form is (4x - 9)^2.

Step by step solution

01

Identify the structure of the trinomial

Recognize that the given trinomial is a perfect square trinomial. A perfect square trinomial has the form a^2 - 2ab + b^2. Attempt to write the given expression in this form.
02

Calculate the square roots

Identify the terms a and b by comparing 16x^2 and 81 with a^2 and b^2 respectively. The square root of 16x^2 is 4x, and the square root of 81 is 9.
03

Verify middle term

Check if the middle term, -72x, can be written as -2ab. Using a = 4x and b = 9, calculate -2ab: -2 (4x)(9) = -72x . It matches the middle term of the polynomial.
04

Factor using the perfect square form

Rewrite 16x^2 - 72x + 81 as (4x - 9)^2 since it is a perfect square trinomial in the form of a^2 - 2ab + b^2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

headline of the respective core concept
Perfect square trinomials are a special type of polynomial. They look like this: \(a^2 - 2ab + b^2\). This special structure makes them easy to factor. Let's see how you can recognize these patterns easily.A trinomial is a perfect square trinomial if:
  • It has three terms.
  • The first and last terms are perfect squares.
  • The middle term is two times the product of the square roots of the first and last terms.
For example: \(16x^2 - 72x + 81\) is a perfect square trinomial. Here's why:
  • The first term \(16x^2\) is \( (4x)^2 \)
  • The last term \ 81 \ is \( 9^2\)
  • The middle term is \( -2ab\). Here, \ a = 4x\ and \b = 9\
headline of the respective core concept
Calculating the square roots of the first and last terms is essential. This is how you find 'a' and 'b' in our pattern. Let's find these roots for \(16x^2 - 72x + 81\):
  • The square root of \ 16x^2 \ is \ 4x \ since \ (4x)^2 = 16x^2 \.
  • The square root of \ 81 \ is \ 9 \ since \ 9^2 = 81 \.
Having these square roots \(a = 4x\ and \b = 9\) helps us verify the middle term and eventually factor the trinomial.
headline of the respective core concept
Verifying the middle term ensures our calculation is correct. To do this, we use the expression \( -2ab \):
  • Given: \ a = 4x\ and \ b = 9\
  • Calculate: \ -2ab = -2 (4x)(9) = -72x \
This matches our middle term! Therefore, we can confidently factor the trinomial as \ (4x - 9)^2 \ since it follows the form \a^2 - 2ab + b^2\.

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Most popular questions from this chapter

In the following exercises, factor using the 'ac' method. $$ 5 n^{2}+21 n+4 $$

In the following exercises, factor using the 'ac' method. $$ 6 y^{2}+y-15 $$

In the following exercises, factor using the 'ac' method. $$ 5 s^{2}-9 s+4 $$

In the following exercises, factor each trinomial of the form \(x^{2}+b x+c .\) $$ q^{2}-13 q+36 $$

In the following exercises, factor each expression using any method. $$ 4 a^{2}+5 a+2 $$
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