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Chapter 5: Problem 115

Factor each polynomial completely. $$a^{2}-16$$

Short Answer

Expert verified
The factored form is \((a+4)(a-4)\).

Step by step solution

01

Identify the polynomial

Recognize the given polynomial: \[a^{2}-16\]
02

Recognize the form

Notice that the polynomial \(a^{2}-16\) is a difference of squares. A difference of squares takes the form \(x^{2} - y^{2}\).
03

Write as a difference of squares

Express \(a^{2}-16\) as \(a^{2} - 4^{2}\). Here, \(x = a\) and \(y = 4\).
04

Apply the difference of squares formula

Use the difference of squares formula, which states \(x^{2} - y^{2} = (x+y)(x-y)\). Applying this to \(a^{2} - 4^{2}\), we get: \[(a + 4)(a - 4)\]

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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 specific type of polynomial expression. It looks like this: \[ x^{2} - y^{2} \] There are two parts: one square term subtracted from another square term. In our exercise, we have \[ a^{2} - 16 \] Notice that 16 is a perfect square (it is \[ 4^{2} \]). So we can rewrite this as: \[ a^{2} - 4^{2} \] This helps us to use the 'difference of squares' formula, which states: \[ x^{2} - y^{2} = (x + y)(x - y) \] Once we substitute \[ a \] for \[ x \] and 4 for \[ y \], it becomes: \[ (a + 4)(a - 4) \].
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations. Here are few examples:
  • \[3x + 2 \]
  • \[5a^{2} - 7b \]
  • \[a^{2} - 16 \]
Expressions don't have an equals sign. They can be simple, like \[ x + 3 \], or more complex, like polynomials. In our exercise, we deal with a special type of algebraic expression 鈥� a polynomial 鈥� which is specifically \[ a^{2} - 16 \]. Understanding algebraic expressions helps simplify and solve problems.
Polynomial Factorization
Polynomial factorization involves breaking down a polynomial into simpler polynomials whose product equals the original. For example: \[a^{2} - 16 \] can be factored using the difference of squares method. Recognizing that \[a^{2} - 4^{2} \] fits the pattern \[x^{2} - y^{2} \], we apply the formula: \[(x + y)(x - y) \] to get: \[ (a + 4)(a - 4) \]. This is now factored completely. Remember, the key steps are:
  • Identify the polynomial form.
  • Recognize the factoring method, like difference of squares.
  • Apply the appropriate formula.
Practicing these steps will make polynomial factorization easier to understand and perform.

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Most popular questions from this chapter

Factor each polynomial using the trial-and-error method. $$ 2 w^{2}+21 w-11 $$

Factor each polynomial by grouping. $$ 6 t-3 t y+a w y-2 a w $$

Solve each equation for \(y .\) Assume a and b are positive numbers. $$a y^{2}+3 y-a y=3$$

Factor each polynomial using the trial-and-error method. $$ 5 y^{2}-14 y-3 $$

On an exam, a student factored \(2 x^{2}-6 x+4\) as \((2 x-4)(x-1) .\) Even though the student carefully checked that $$ (2 x-4)(x-1)=2 x^{2}-6 x+4 $$ the student lost some points. What went wrong?
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