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Chapter 5: Problem 60

Factor completely. \(a^{2}-8 a+16-b^{2}\)

Short Answer

Expert verified
(a - 4 + b)(a - 4 - b)

Step by step solution

01

Recognize the structure of the polynomial

Notice that the given polynomial is in the form of a difference of squares: a^2 - 8a + 16 - b^2.The expression can be reorganized for easier factoring.
02

Identify and rewrite the perfect square trinomial

Observe that the first part of the expression, a^2 - 8a + 16,is a perfect square trinomial. It can be rewritten as,(a - 4)^2.
03

Express the polynomial as a difference of squares

We now have an expression: (a - 4)^2 - b^2.This is a classic difference of squares, which can be factored further using the formula x^2 - y^2 = (x + y)(x - y).
04

Apply the difference of squares formula

Using the formula for the difference of squares to factor, we get:(a - 4)^2 - b^2 = ( (a - 4) + b )( (a - 4) - b ).

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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 type of polynomial. It takes the form of \(a^2 - b^2\). Here, both terms are perfect squares. For example, in the expression \((a - 4)^2 - b^2\), \(a - 4\) and \(b\) are squared terms.
This pattern can always be factored into two binomials as follows: \(a^2 - b^2 = (a + b)(a - b)\).
Understanding this formula helps simplify and solve polynomial expressions efficiently.
Identifying and using the difference of squares can turn seemingly complex polynomials into simple product forms.
perfect square trinomial
A perfect square trinomial is a polynomial that can be written as the square of a binomial. It generally takes the form \(a^2 \text{-} 2ab \text{+} b^2 = (a - b)^2\).
In the given exercise, we have \(a^2 - 8a + 16\), where:
  • \(a^2\) is the square of \(a\),
  • \(16\) is the square of \(4\),
  • \(-8a\) is twice the product of \(a\) and \(4\).
It's crucial to recognize these patterns. Rewriting the trinomial as \((a - 4)^2\) clarifies the structure, making further factoring possible.
polynomial factoring steps
Factoring polynomials involves breaking down a complex expression into simpler components. The steps taken in the given exercise are useful for various problems:
  • Step 1: Recognize the structure: Identify the form of the polynomial. For example, whether it's a simple binomial, trinomial, or special product form.
  • Step 2: Reorganize: Rewrite the polynomial if necessary. This helps in recognizing familiar patterns.
  • Step 3: Identify specific forms: Look for perfect square trinomials or the difference of squares.
  • Step 4: Apply relevant formulas: Use factoring formulas like \((a \text{卤} b)^2 = a^2 \text{卤} 2ab \text{+} b^2\) or \(a^2 - b^2 = (a + b)(a - b)\).
Practicing these steps improves problem-solving skills and simplifies complex polynomials effectively.

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