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Chapter 6: Problem 45

In the following exercises, factor by grouping. $$ 9 p^{2}+12 p-15 p-20 $$

Short Answer

Expert verified
The factored expression is (3p + 4)(3p - 5).

Step by step solution

01

- Group Terms

First, group the terms into two pairs. The expression can be grouped as i) (9p^2 + 12p)ii) (-15p - 20)
02

- Factor Out the Greatest Common Factor (GCF) from Each Pair

From the first pair (9p^2 + 12p),the greatest common factor is 3p. Factoring out 3p gives us: 3p(3p + 4).From the second pair (-15p - 20),the greatest common factor is -5. Factoring out -5 gives us: -5(3p + 4).
03

- Factor Out the Common Binomial

Now the expression looks like this: 3p(3p + 4) - 5(3p + 4). Notice that (3p + 4) is a common factor in both terms. Factor this binomial out: (3p + 4)(3p - 5).

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Key Concepts

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

factoring polynomials
Factoring polynomials is a useful technique in algebra that allows us to express a polynomial as the product of simpler polynomials. This can simplify solving equations and finding roots. The main steps for factoring are:
  • Identifying and grouping terms.
  • Finding the greatest common factor (GCF) for each group.
  • Factoring out the common binomial.
Always remember to double-check your work by expanding the factored form to see if you get back to the original polynomial.
greatest common factor
The greatest common factor (GCF) is the largest factor that divides two or more numbers. When factoring in algebra, we look for the GCF of terms to simplify expressions. For example:
  • In the pair \(9p^2 + 12p\), the GCF is \(3p\).
  • In the pair \(-15p - 20\), the GCF is \(-5\).
Factoring out the GCF simplifies the expression and makes it easier to find common factors between terms.
algebraic expressions
Algebraic expressions include variables, numbers, and arithmetic operations. Understanding how to manipulate these expressions is crucial in algebra. Key steps include:
  • Recognizing like terms and grouping them together.
  • Applying operations like addition, subtraction, and factoring.
For example, in the polynomial \(9p^2 + 12p - 15p - 20\), recognizing common factors and grouping them correctly allowed us to factor it into \((3p + 4)(3p - 5)\). This simplification reveals important properties of the expression and can help solve related problems.

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Most popular questions from this chapter

In the following exercises, factor. $$ -18 x y^{2}-27 x^{2} y $$

In the following exercises, factor using the 'ac' method. $$ 4 k^{2}-16 k+15 $$

In the following exercises, factor each trinomial of the form \(x^{2}+b x+c .\) $$ m^{2}-13 m+30 $$

In the following exercises, factor using the 'ac' method. $$ 30 x^{2}+105 x-60 $$

In the following exercises, factor by grouping. $$ x^{2}-x+4 x-4 $$
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