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Chapter 6: Problem 45

For Problems \(11-46\), factor each polynomial completely. (Objective 2) $$ 2 x(x+1)-3(x+1) $$

Short Answer

Expert verified
The expression factors to \((x+1)(2x-3)\).

Step by step solution

01

Identify Common Factors

First, look for any common factors in the expression \(2x(x+1) - 3(x+1)\). Notice that both terms include \((x+1)\) as a factor.
02

Factor Out Common Expression

Factor \((x+1)\) out of the polynomial: \((x+1)(2x - 3)\). This step involves factoring the whole expression by recognizing \((x+1)\) as the greatest common factor of the terms.
03

Verify the Factoring

To ensure the polynomial is correctly factored, distribute \((x+1)\) back into \((2x-3)\). This should reproduce the original polynomial: \((x+1)(2x) + (x+1)(-3) = 2x(x+1) - 3(x+1)\). Confirming this verifies the factoring is correct.

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Key Concepts

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

Greatest Common Factor
The greatest common factor (GCF) is a crucial concept in factoring polynomials. It's like finding the biggest piece that fits into multiple different parts of a puzzle. When you're working with polynomials, the GCF is the largest expression that can evenly divide each term in the polynomial.
For example, in the expression \(2x(x+1) - 3(x+1)\), the factor \((x+1)\) is common to both terms. Identifying this common factor is the first step in factoring the polynomial.
  • Look for terms or expressions that repeat across different parts of the polynomial.
  • The GCF is not just found with numbers; it can be a whole expression.
Once the GCF is identified, it can be factored out to simplify the polynomial, making further algebraic manipulation easier.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operators like addition and subtraction. In order to factor polynomials, you need to be comfortable working with these expressions.
In the polynomial \(2x(x+1) - 3(x+1)\), you see both multiplication and addition. The task of factoring involves simplifying this expression by recognizing and pulling out common elements or structures.
  • Each part of the expression is important: coefficients like '2' and '-3' and expressions like \((x+1)\).
  • Understanding the structure helps in breaking down, or "factoring," the expression into simpler parts.
Getting comfortable with manipulating algebraic expressions is vital for tasks like combining like terms or distributing factors.
Verify Factoring
Verification is the process of checking your work to make sure it's correct. After you factor a polynomial, it's essential to go back and confirm that the expression can be multiplied out to the original form.
Consider the factored form \((x+1)(2x - 3)\). By distributing, or multiplying, \((x+1)\) back into \((2x - 3)\), you should end up with the original expression \(2x(x+1) - 3(x+1)\).
  • Distribute each term in the first factor by every term in the second.
  • Compare the result with the original polynomial to ensure they match.
Verification reduces errors and increases confidence in your factoring work.

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Most popular questions from this chapter

For Problems \(71-88\), set up an equation and solve each problem. (Objective 4) Suppose that the combined area of two squares is 360 square feet. Each side of the larger square is three times as long as a side of the smaller square. How big is each square?

For Problems \(13-40\), factor each polynomial completely. Indicate any that are not factorable using integers. (Objective 2) $$ x^{2}+12 x y+36 y^{2} $$

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ x^{3}=-4 x^{2} $$

The length of one leg of a right triangle is 3 centimeters more than the length of the other leg. The length of the hypotenuse is 15 centimeters. Find the lengths of the two legs.

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ x^{3}+10 x^{2}+25 x=0 $$
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