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Chapter 5: Problem 131

Factor each polynomial completely. \(9 a^{2}-48 a b+64 b^{2}-4 c^{2}\)

Short Answer

Expert verified
The complete factorization is \((3a-8b-2c)(3a-8b+2c)\).

Step by step solution

01

- Identify Quadratic Form

The given polynomial is in the quadratic form: \[ 9a^{2} - 48ab + 64b^{2} - 4c^{2} \]
02

- Group Terms

Group the first three terms and the last term separately: \[ 9a^{2} - 48ab + 64b^{2} - 4c^{2} \]
03

- Identify a Common Pattern

Notice that the first three terms \(9a^2 - 48ab + 64b^2\) can be written as a perfect square. Rewrite it: \[ (3a - 8b)^2 - 4c^2 \]
04

- Apply the Difference of Squares Formula

Recognize the expression as a difference of squares: \[ (3a - 8b)^2 - (2c)^2 \]. Use the formula for difference of squares \(x^2 - y^2 = (x-y)(x+y)\): \[ (3a - 8b - 2c)(3a - 8b + 2c) \]

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Key Concepts

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

Difference of Squares
The difference of squares is a way to factor certain types of polynomials. Essentially, it applies to expressions that look like this: a square number minus another square number. Notice that this fits the pattern: \[ x^2 - y^2 \] You can factor this easily using: \[ (x - y)(x + y) \] For example, consider \[ (3a - 8b)^2 - (2c)^2 \]. This fits our pattern, where \( x = 3a - 8b \) and \( y = 2c \). So, it factors to: \[ (3a - 8b - 2c)(3a - 8b + 2c) \].
Perfect Square Trinomial
A perfect square trinomial is another type of polynomial that is easy to factor. Look for an expression in the form \[ a^2 + 2ab + b^2 \] or \[ a^2 - 2ab + b^2 \]. Both of these patterns will factor into a binomial squared. Take \[ 9a^2 - 48ab + 64b^2 \], for instance. It matches the form of \( a^2 - 2ab + b^2 \) where \( a = 3a \) and \( b = 8b \). So, it can be written as: \[ (3a - 8b)^2 \]
Quadratic Form
Quadratic forms appear frequently in polynomial factoring, typically in the form \[ Ax^2 + Bx + C \]. Recognizing this form helps simplify complex expressions intricately. In the exercise \[ 9a^2 - 48ab + 64b^2 - 4c^2 \], the first three terms \[ 9a^2 - 48ab + 64b^2 \] represent a quadratic form. Identifying and factoring this part produces \[ (3a - 8b)^2 \], leading into the next conceptual steps. Recognizing these standard patterns is essential for efficient polynomial factoring.
Grouping Terms
Grouping terms involves organizing a polynomial to simplify factoring. In the given polynomial, \[ 9a^2 - 48ab + 64b^2 - 4c^2 \], you start by identifying parts you can manage easier. Here, you can group the first three terms separately from the last term: \[ 9a^2 - 48ab + 64b^2 \] and \[ -4c^2 \]. Within grouped terms, it's easier to see how to apply patterns, such as perfect square trinomials or the difference of squares. Grouping breaks down complex expressions into manageable chunks.

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Most popular questions from this chapter

Find the value of the indicated variable. Find \(c\) so that \(4 m^{2}-12 m+c\) factors as \((2 m-3)^{2}\).

List all pairs of integers with the given product. Then find the pair whose sum is given. Product: \(48 ; \quad\) Sum: -19

Solve each equation, and check the solutions. $$ (x-1)(x+8)=0 $$

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Factor each binomial completely. \(125 t^{3}+8 s^{3}\)
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