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Chapter 5: Problem 189

Factor each polynomial completely. $$a^{4}-16$$

Short Answer

Expert verified
\(a^4 - 16 = (a - 2)(a + 2)(a^2 + 4)\)

Step by step solution

01

Identify the difference of squares

The expression given is a difference of squares. Recall that the difference of squares formula is: \[x^2 - y^2 = (x - y)(x + y)\]. Here, treat the terms as squares: \[a^4 = (a^2)^2\] and 16 as \[4^2\].
02

Apply the difference of squares formula

Using the formula, the expression \[a^4 - 16\] can be rewritten as: \[(a^2)^2 - 4^2 = (a^2 - 4)(a^2 + 4)\].
03

Factor the difference of squares again

Notice that \[a^2 - 4\] is also a difference of squares: \[a^2 - 4 = (a)^2 - 2^2\]. Applying the formula again: \[a^2 - 4 = (a - 2)(a + 2)\].
04

Combine all factors

Substitute the factors back into the original expression: \[a^4 - 16 = (a^2 - 4)(a^2 + 4)\], and factoring \[a^2 - 4\] gives: \[(a^2 - 4)(a^2 + 4) = (a - 2)(a + 2)(a^2 + 4)\].

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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 special algebraic expression. It is in the form of: \[a^2 - b^2\]This can be factored using the formula:\[a^2 - b^2 = (a - b)(a + b)\]This is an essential pattern in algebra. It simplifies polynomials and makes solving them easier. Understanding this concept helps to recognize similar patterns in more complex expressions.
algebraic expressions
Algebraic expressions are combinations of constants and variables. They include operations like addition, subtraction, multiplication, and division. For example, \[3x + 2\] and \[a^2 - 4\] are algebraic expressions. In the expression \[a^4 - 16\], we work with variables raised to a power. Factoring such expressions requires recognizing specific patterns, such as the difference of squares, which simplifies the process.
polynomial factorization
Polynomial factorization is the process of breaking down a polynomial into simpler factors. These factors, when multiplied together, give the original polynomial. For the exercise \[a^4 - 16\], the steps were:
  • Identify the difference of squares: Recognize \[a^4 - 16 = (a^2)^2 - 4^2\]
  • Apply the difference of squares formula:\[ (a^2 - 4)(a^2 + 4)\]
  • Factor further: Recognize \[a^2 - 4 = (a - 2)(a + 2)\]
  • Combine all factors: \[ (a - 2)(a + 2)(a^2 + 4)\]
Understanding these steps ensures you can factor any polynomial effectively, revealing simpler forms useful for solving equations and other algebraic applications.

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Most popular questions from this chapter

Factor each polynomial using the trial-and-error method. $$ m^{2}+8 m-9 $$

Solve each equation for \(y .\) Assume a and b are positive numbers. $$y^{2}-b^{2}=0$$

Factor each polynomial by grouping. $$ y^{2}-a y-b y+a b $$

Solve each equation. $$(3 x-5)\left(25 x^{2}-4\right)=0$$

Factor each polynomial using the trial-and-error method. $$ a^{2}-9 a+18 $$
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