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Chapter 6: Problem 41

For Problems \(11-46\), factor each polynomial completely. (Objective 2) $$ a(b-4)-c(b-4) $$

Short Answer

Expert verified
The polynomial factors completely to \((b-4)(a-c)\).

Step by step solution

01

Identify the Common Factor

Look for a common term in each part of the expression. In the given polynomial, the term \((b-4)\) is common in both parts: \(a(b-4)\) and \(-c(b-4)\). This can be factored out to simplify the expression.
02

Factor Out the Common Term

Factor \((b-4)\) out of the expression. This gives: \( (b-4)(a) - (b-4)(c) = (b-4)(a-c) \).
03

Write the Final Factored Form

After factoring \((b-4)\) from both terms, the polynomial is completely factored as: \( (b-4)(a-c) \).

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Key Concepts

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

Common Factors
Identifying common factors is a vital step in simplifying polynomial expressions. It involves spotting terms that appear in each part of the expression. In this exercise, the term \(b-4\) is present in both \(a(b-4)\) and \(-c(b-4)\). Recognizing these common terms allows us to factor them out, making the expression more manageable and preparing it for further simplification or solution.
  • First, look at each part of the polynomial separately.
  • Identify terms that are repeated across the entire expression.
  • Factor them out to simplify the polynomial.
Breaking a complex problem into simpler parts by recognizing common factors is a valuable math skill. It sets the stage for making the problem much more straightforward to solve.
Factored Form
Once common factors have been identified, the next step is to convert the expression into its factored form. Factored form is a way of simplifying an expression into a product of simpler expressions. In our example, the original expression \(a(b-4) - c(b-4)\) is transformed by factoring out \(b-4\), resulting in the factored form \( (b-4)(a-c) \).
  • This step involves writing the expression as a product of its factors.
  • Ensure all terms are accounted for when rewriting the expression.
  • Factored expressions are often easier to work with, especially when solving equations.
The factored form provides a clear and concise representation of the polynomial, which can reveal solutions more easily or simplify further computations.
Polynomial Expressions
Polynomial expressions are mathematical phrases that can include constants, variables, and exponents. They are defined as the sum of terms that have non-negative integer exponents. The expression we dealt with is a type of polynomial, consisting of two terms: \(a(b-4)\) and \(-c(b-4)\).
  • Polynomials can include one or multiple terms combined by addition or subtraction.
  • These expressions are pivotal in algebra and calculus, forming the backbone of functions and equations.
  • Understanding polynomials involves recognizing their structure and applying operations like factoring to manipulate them.
By mastering polynomial expressions and operations such as factoring, you enhance your math toolkit, aiding you in handling more complex algebraic problems efficiently.

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Most popular questions from this chapter

Explain how you would solve \((4 x-3)(8 x+5)=0\) and also how you would solve \((4 x-3)(8 x+5)=-9\).

\(n^{2}+5 n-36=0\)

\(x^{2}+10 x+21=0\)

For Problems \(71-88\), set up an equation and solve each problem. (Objective 4) The area of a triangle is 51 square inches. One side of the triangle is 1 inch less than three times the length of the altitude to that side. Find the length of that side and the length of the altitude to that side.

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ 24 x^{2}+74 x-35=0 $$
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