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Chapter 6: Problem 5

Factor completely. If a polynomial is prime, state this. $$ 10 a^{2}-640 $$

Short Answer

Expert verified
10(a + 8)(a - 8)

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First, observe the polynomial and identify the greatest common factor (GCF) of each term. Here, the terms are 10 and 640. The GCF is 10.
02

Factor Out the GCF

Divide each term by the GCF and factor it out from the polynomial: \[10a^2 - 640 = 10(a^2 - 64)\]
03

Recognize and Apply the Difference of Squares

Notice that \[a^2 - 64\] is a difference of squares. The difference of squares formula is \(x^2 - y^2 = (x + y)(x - y)\). Here, apply this formula with \(a^2 - 64 = (a + 8)(a - 8)\).
04

Combine All Factors

Combine the extracted GCF with the factored form from the difference of squares: \[10(a + 8)(a - 8)\]

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Key Concepts

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

Greatest Common Factor
When factoring polynomials, it's crucial to start by finding the Greatest Common Factor (GCF). The GCF is the largest number that can divide each term in the polynomial without leaving a remainder.
For example, consider the expression in this exercise:
  • Terms: 10 and 640
  • The GCF of 10 and 640 is 10
To factor out the GCF, you divide each term by this common factor and factor it out in front of a parenthesis. Hence,
The expression
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Difference of Squares
The difference of squares is a specific technique used in polynomial factoring, particularly when your polynomial appears in the form of two squared terms separated by a subtraction sign.
This technique relies on the formula:
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Most popular questions from this chapter

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 8 x^{2}-8 y^{2} $$

Solve. Round any irrational solutions to the nearest thousandth. $$ x^{3}+3 x^{2}+x-1=0 $$

Find the zeros of each function. $$ f(x)=2 x^{2}+4 x+2 $$

Factor completely. Assume that variables in exponents represent positive integers. $$ 0.09 x^{8}+0.48 x^{4}+0.64 $$

Solve. Round any irrational solutions to the nearest thousandth. $$ x^{2}=\frac{1}{100} $$
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