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Chapter 10: Problem 45

Factor out the greatest common factor. $$3 x^{3}+5 x^{2}+7 x$$

Short Answer

Expert verified
The factored form is \(x(3x^2 + 5x + 7)\).

Step by step solution

01

Identify the common factors

Examine each term in the polynomial to identify common factors. The terms are: \(3x^3\), \(5x^2\), and \(7x\). Each term contains a factor of \(x\).
02

Determine the greatest common factor (GCF)

The GCF of the terms in the polynomial \(3x^3\), \(5x^2\), and \(7x\) is \(x\).
03

Factor out the GCF

Rewrite each term in the polynomial by factoring out the GCF, \(x\): \[ 3x^3 + 5x^2 + 7x = x(3x^2) + x(5x) + x(7)\]
04

Simplify the expression

Combine the factored terms: \[ 3x^3 + 5x^2 + 7x = x(3x^2 + 5x + 7) \]

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Key Concepts

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

greatest common factor
When factoring polynomials, finding the greatest common factor (GCF) is a crucial step. The GCF is the highest number or variable that can evenly divide each term in the polynomial. For instance, in the polynomial given: \(3x^3 + 5x^2 + 7x \), each term shares a common factor of \(x\). Here, \(x\) is the factor present in all terms, so it is our GCF. Determining the GCF helps in simplifying and reducing polynomial expressions efficiently.
polynomial expression
A polynomial is an expression made up of variables, coefficients, and exponents. It consists of terms added or subtracted together. The polynomial in our exercise is: \(3x^3 + 5x^2 + 7x \). Each term could stand alone as a monomial, but when combined, they form a polynomial. Understanding these components is key to mastering polynomial manipulation.
  • \textbf{Coefficients}: The numbers in front of the variables (3, 5, 7).
  • \textbf{Variables}: The letters representing unknown values (x in this case).
  • \textbf{Exponents}: The numbers indicating how many times to multiply the variable (3, 2, 1).
Recognizing these elements helps us systematically approach problems involving polynomials.
factoring step-by-step
Factoring polynomials is about breaking down a polynomial into simpler terms or products. Here are the steps using the given polynomial:
  • \textbf{Step 1: Identify the common factors.}
    Examine each term: \(3x^3, 5x^2, 7x \). Notice that each term contains \(x\).
  • \textbf{Step 2: Determine the greatest common factor (GCF).}
    Since \(x\) is common in all terms, \(x\) becomes the GCF.
  • \textbf{Step 3: Factor out the GCF.}
    Rewrite the polynomial by factoring out \(x\):

    \[ 3x^3 + 5x^2 + 7x = x(3x^2) + x(5x) + x(7) \]
  • \textbf{Step 4: Simplify the expression.}
    Combine the remaining terms within the parenthesis: \[ 3x^3 + 5x^2 + 7x = x(3x^2 + 5x + 7) \]

By following these steps, you can easily factor polynomials and simplify them to a more manageable form.

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Most popular questions from this chapter

Subtract the polynomials. $$\left(2 m n^{3}+6 m^{2} n^{2}+9 m n^{2}-3 m n\right)-\left(5 m n^{3}-2 m^{2} n^{2}-7 m n\right)$$

Write a binomial with degree 4.

Simplify. Write the answers with positive exponents only. $$z^{-4} z^{7}$$

The cost (in dollars) for a speeding ticket is given by the polynomial \(110+15 x .\) In this context, \(x\) is the number of miles per hour a motorist travels over the speed limit. a. Evaluate the polynomial for \(x=15\) and interpret the answer in the context of this problem. b. Evaluate the polynomial for \(x=25\) and interpret the answer in the context of this problem.

Subtract the polynomials. $$\begin{array}{r}5 x^{2}+9 x-6 \\\\-\left(3 x^{2}-2 x+1\right) \\\\\hline\end{array}$$
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