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Chapter 5: Problem 83

Factor each polynomial completely. $$ 2 x^{2}-128 y^{2} $$

Short Answer

Expert verified
2(x + 8y)(x - 8y)

Step by step solution

01

Identify common factors

First, look for any common factors in the polynomial. In this case, the common factor is 2.
02

Divide by the common factor

Divide each term in the polynomial by the common factor. This gives: 2(x^2 鈭� 64y^2)
03

Recognize the difference of squares

Next, notice that the expression inside the parentheses is a difference of squares: ( x^2 鈭� 64y^2 ).
04

Factor the difference of squares

A difference of squares can be factored into two binomials: (a^2 - b^2) = (a + b)(a - b). Apply this to the polynomial: x^2 鈭� 64y^2 = (x + 8y)(x 鈭� 8y)
05

Combine all factors

Put the common factor 2 back in front of the factored form: 2(x^2 鈭� 64y^2) = 2(x + 8y)(x 鈭� 8y).

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Key Concepts

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

Understanding Common Factors
Common factors are numbers or variables that are present in every term of a polynomial. Identifying these common factors is an essential first step in simplifying or factoring polynomials.
  • For example, in the polynomial \(2 x^{2}-128 y^{2}\), the number 2 is a common factor in both terms.

How to Find Common Factors:
  • First, look at each term in the polynomial separately.
  • Identify any numbers and/or variables that appear in all the terms.
  • Factor out the greatest common factor.

Factoring out common factors simplifies the polynomial and prepares it for further factoring steps.
The Difference of Squares
The difference of squares is a special factoring technique that applies to expressions of the form \(a^2 - b^2\). This expression can always be factored into the product \((a + b)(a - b)\).
  • Recognize the difference of squares: an expression where two perfect squares are subtracted.

Identifying Perfect Squares:
  • Analyze whether each term in the subtraction is a perfect square, such as \(x^2\) or \(64y^2\).
  • If yes, express each term in its squared form.

Applying the Formula:
  • Use the formula \((a^2 - b^2) = (a + b)(a - b)\) to factor the expression.
In our specific example, \(x^2\) and \(64y^2\) make use of the factorization: \((x + 8y)(x 鈭� 8y)\).
Understanding Binomials
A binomial is a polynomial with exactly two terms. Both terms can contain coefficients, variables, or both.
  • For example, \((x + 8y)\) and \((x - 8y)\) are binomials.

Important Components of Binomials:
  • Each binomial has two terms separated by either a plus (+) or minus (鈭�) sign.
  • Factoring binomials involves combinations of addition and subtraction that form within polynomial equations.
When factoring the difference of squares like in our example, the result typically comes out as the product of two binomials.

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