Problem 94 Use a pattern to factor. Identif... [FREE SOLUTION]

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Elementary Algebra

Laura Bracken, Ed Miller

$Math Studyset 91影视 Explanations$ Math

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Chapter 8: Problem 94

Use a pattern to factor. Identify any prime polynomials. $$9 a^{2}-b^{2}$$

Short Answer

Expert verified

The factored form is $ (3a + b)(3a - b) $.

Step by step solution

- Recognize the Difference of Squares

The given expression is of the form $ A^{2} - B^{2} $. Notice that both terms are perfect squares. Here, \[ A^{2} = 9a^{2} \text{ and } B^{2} = b^{2} \].

- Find the Square Roots

Identify the square roots of the terms. The square root of $ 9a^{2} $ is $ 3a $. The square root of $ b^{2} $ is $ b $. Thus, \[ A = 3a \text{ and } B = b \].

- Apply the Difference of Squares Formula

Using the difference of squares formula \[ A^2 - B^2 = (A+B)(A-B) \], substitute $ A = 3a $ and $ B = b $ into the formula:\[ 9a^2 - b^2 = (3a + b)(3a - b) \].

- Check for Prime Polynomials

Examine the factors $ (3a + b) $ and $ (3a - b) $. Since neither of these can be factored further using integer coefficients, they are considered prime polynomials.

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Key Concepts

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

Difference of Squares

The concept of 'Difference of Squares' is critical in algebra. It applies when you subtract one perfect square from another, represented as $ A^2 - B^2 $. This formula is special because it can be factored into two binomials. Here's the formula: \[ A^2 - B^2 = (A + B)(A - B) \]. Recognizing this pattern helps immensely in simplifying and solving polynomial equations. For example, with $ 9a^2 - b^2 $, you identify $ A = 3a $ and $ B = b $. Hence, \[ 9a^2 - b^2 = (3a + b)(3a - b) \].

Perfect Squares

Perfect squares are numbers or expressions that are the square of an integer or polynomial. For instance, $ 9a^2 $ and $ b^2 $ are perfect squares because $ 9a^2 $ is $ (3a)^2 $ and $ b^2 $ is $ b^2 $. Recognizing perfect squares can help you factor expressions more easily. When both terms in a binomial are perfect squares, you might be dealing with a difference of squares. This recognition leads to easier factorization using patterns.

Prime Polynomials

A prime polynomial cannot be factored into products of polynomials with integer coefficients. After factoring an expression, you should check if further factorization is possible. In the example $ (3a + b)(3a - b) $, neither binomial can be factored further with integers. Thus, they are prime polynomials. Knowing when a polynomial is prime saves time and effort in solving algebraic problems.

Factoring Techniques

Factoring techniques are essential tools in algebra. These include methods like recognizing patterns (e.g., difference of squares), grouping, or using formulas (e.g., \[ A^2 - B^2 = (A + B)(A - B) \]). The step-by-step approach is vital:

Identify patterns (e.g., perfect squares, difference of squares)
Find the square roots of terms
Apply the appropriate factoring formula
Check if the resulting factors are prime

Practicing these techniques will improve your algebra problem-solving skills.

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91影视

Use a pattern to factor. Identify any prime polynomials. $$9 a^{2}-b^{2}$$

Short Answer

Step by step solution

- Recognize the Difference of Squares

- Find the Square Roots

- Apply the Difference of Squares Formula

- Check for Prime Polynomials

Key Concepts

Difference of Squares

Perfect Squares

Prime Polynomials

Factoring Techniques

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