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Chapter 6: Problem 9

In the following exercises, factor the greatest common factor from each polynomial. $$ 6 m+9 $$

Short Answer

Expert verified
\(3(2m + 3)\)

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First, find the greatest common factor (GCF) of the coefficients. The coefficients are 6 and 9. The largest number that evenly divides both 6 and 9 is 3.
02

Rewrite Each Term with the GCF

Rewrite each term as a product of the GCF and another factor. For the term 6m, it can be written as \(3 \times 2m\). For the term 9, it can be written as \(3 \times 3\).
03

Factor Out the GCF

Factor out the GCF (3) from each term in the polynomial. This gives \(3(2m + 3)\).

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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 the largest number that can evenly divide each of the given terms of a polynomial. Finding the GCF is crucial for simplifying expressions and for polynomial factorization.

For example, consider the polynomial given: \(6m + 9\). The coefficients here are 6 and 9.

To find the GCF:
  • List the factors of each number:
    • 6: 1, 2, 3, 6
    • 9: 1, 3, 9
  • Identify the largest common factor:
    In this case, it's 3.
Once you identify the GCF, you can use it to rewrite and simplify the polynomial.
polynomial factorization
Polynomial factorization involves expressing a polynomial as a product of its factors. It simplifies complex expressions, making them easier to work with.

Here's the process, using the polynomial \(6m + 9\):
  • First, find the GCF: In this case, it is 3.
  • Next, rewrite each term of the polynomial as a product of the GCF and another factor:
    • \(6m\) becomes \(3 \times 2m\)
    • \(9\) becomes \(3 \times 3\)
Now, factor out the GCF from each term. This yields:
\(3(2m + 3)\). Now, the polynomial is factored down to the simplest form involving its GCF.
algebraic expressions
Algebraic expressions are mathematical phrases that can contain numbers, variables, and arithmetic operations. Polynomials are a specific type of algebraic expression.

In the polynomial \(6m + 9\), we have two terms:
  • \(6m\): which is a product of 6 and \(m\)
  • 9: a constant term
Each term in an algebraic expression contributes to the overall use and function of the expression in a given problem.

By factoring an algebraic expression like a polynomial, we simplify the equation and make it easier to manipulate and solve.
mathematical coefficients
In polynomials, coefficients are the numerical part of the terms involving variables. They indicate how many times the variable is multiplied.

For the polynomial \(6m + 9\), the coefficients are:
  • 6: in the term \(6m\)
  • 9: in the constant term 9
Understanding coefficients is essential when factoring polynomials since they play a crucial role in identifying the GCF.

By focusing on the coefficients, we can break down complex expressions into factor groups, simplifying our calculations and the overall problem-solving process.

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Most popular questions from this chapter

Solve. Gianna is going to throw a ball from the top floor of her middle school. When she throws the ball from 48 feet above the ground, the function \(h(t)=-16 t^{2}+32 t+48\) models the height, \(h\), of the ball above the ground as a function of time, \(t\). Find: (a) the zeros of this function which tells us when the ball will hit the ground. (b) the time(s) the ball will be 48 feet above the ground. (c) the height the ball will be at \(t=1\) seconds which is when the ball will be at its highest point.

In the following exercises, factor each trinomial of the form \(x^{2}+b x+c .\) $$ y^{2}-18 y+45 $$

In the following exercises, factor completely using trial and error. $$ 4 a^{2}+17 a b-15 b^{2} $$

Factor completely using the sums and differences of cubes pattern, if possible. $$ x^{3}+125 $$

What does it mean to say a polynomial is in factored form?
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