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Chapter 5: Problem 112

Factor each polynomial completely. $$b^{2}+6 b+9$$

Short Answer

Expert verified
The polynomial can be factored as

Step by step solution

01

Identify the Polynomial

The given polynomial is a quadratic expression:
02

Recognize the Perfect Square

Notice that the given polynomial can be rewritten in a specific format if it is a perfect square trinomial. For a trinomial of the form
03

Rewrite in Perfect Square Form

Express the coefficient of the middle term as twice the product of a single variable and another term. Confirm that it satisfies the conditions for a perfect square trinomial. Rewrite the given expression in the standard format:
04

Write as a Binomial Squared

Upon confirming that the conditions for a perfect square are met, write the polynomial as the square of a binomial:

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Key Concepts

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

perfect square trinomial
A perfect square trinomial is a special type of quadratic expression. It takes the form \(a^2 + 2ab + b^2\), where:
  • \(a^2\) is the square of the first term.
  • \(2ab\) is twice the product of the two terms.
  • \(b^2\) is the square of the second term.
To recognize a perfect square trinomial, compare your expression to this format.
If it matches, you can factor it into a binomial squared seamlessly.
quadratic expression
A quadratic expression is generally written in the form \(ax^2 + bx + c\), where
  • \(a\), \(b\), and \(c\) are constants.
  • \(x\) is the variable.
One common task in math is to factor these expressions.
When recognizing patterns, you may identify some quadratic expressions as perfect square trinomials. This makes the factoring process straightforward.
Quadratic expressions can describe parabolic graphs, which appear in many applied mathematics contexts.
binomial squared
A binomial squared is an expression in the form \((a + b)^2\).
When expanded, it results in a perfect square trinomial: \(a^2 + 2ab + b^2\). Here's a step-by-step to convert a perfect square trinomial back to binomial squared:
  • Identify the terms corresponding to \(a^2\) and \(b^2\).
  • Check that the middle term equals \(2ab\).
  • Rewrite the expression using the binomial squared format.
For instance: Given \(b^2 + 6b + 9\), we can see:
  • \(a = b\)
  • \(b = 3\)
  • The middle term \(= 2 \cdot b \cdot 3\)
This confirms it can be written as \((b+3)^2\).

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Most popular questions from this chapter

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