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Chapter 7: Problem 24

Factor the greatest common factor from each polynomial. \(7 r+7\)

Short Answer

Expert verified
7(r + 1)

Step by step solution

01

Identify the Common Factor

Examine each term in the polynomial to determine the greatest common factor (GCF). The terms are \(7r\) and \(7\). The GCF of these terms is \(7\).
02

Factor Out the GCF

Express the polynomial as a product of the GCF and the remaining factors. Factor out \(7\) from each term: \(7(r + 1)\).
03

Write the Final Factored Form

Combine the GCF and the remaining polynomial to write the final factored form: \(7(r + 1)\).

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Key Concepts

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

greatest common factor
Understanding the greatest common factor (GCF) is crucial in algebra. The GCF of two or more numbers or terms is the largest factor that all terms share. Finding the GCF helps simplify polynomial expressions.

To identify the GCF of polynomial terms, follow these steps:

  • List the factors of each term. For instance, if you have the terms 7r and 7, list the factors of each. For 7r, the factors are 1, 7, r, and 7r. For 7, the factors are 1 and 7.
  • Identify the largest number that is a factor of all terms. Here, the largest common factor is 7.
Understanding how to find and use the GCF simplifies many algebraic operations and is essential when factoring polynomials.
factoring steps
Properly factoring polynomials involves a series of systematic steps. Applying these steps will ensure accurate results and a deeper understanding of algebra.

Here are the steps to follow when factoring polynomials:

  • Step 1: Identify the Common Factor: Examine all terms in the polynomial to find the greatest common factor (GCF). For example, in the polynomial 7r + 7, the GCF is 7.
  • Step 2: Factor Out the GCF: Once the GCF is found, divide each term in the polynomial by this factor. In 7r + 7, factor out 7 to get 7(r + 1).
  • Step 3: Write the Final Factored Form: Express the polynomial as the product of the GCF and the remaining polynomial. The factored form of 7r + 7 is 7(r + 1).
By diligently following these steps, students can effectively simplify and factor polynomial expressions.
polynomial expressions
Polynomial expressions consist of terms that include variables raised to positive integer powers and coefficients. They appear frequently in algebra and higher-level mathematics.

Here are key points to understand about polynomial expressions:

  • Terms: Each polynomial is made up of terms, which are products of coefficients and variables. In the expression 7r + 7, '7r' and '7' are terms.
  • Degree: The degree of a polynomial is the highest power of the variable in its terms. For instance, the expression 2x^3 + 4x has a degree of 3.
  • Simplification: Simplifying polynomials often involves combining like terms and factoring. Factoring a polynomial by its GCF, like turning 7r + 7 into 7(r + 1), simplifies the expression and reveals its structure.
Understanding these elements and how they interact is essential for mastering polynomials and succeeding in algebra.

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Most popular questions from this chapter

Factor completely. \(5 n^{3}+320\)

The area of a rectangular carpet is 28 square feet. The length is three feet more than the width. Find the length and the width of the carpet.

Maribel factored \(y^{2}-30 y+81\) as \((y-9)^{2}\). Was she right or wrong? How do you know?

The product of two consecutive integers is 42. Find the integers.

A fall tradition at the University of California San Diego is the Pumpkin Drop, where a pumpkin is dropped from the eleventh story of Tioga Hall. (a) The binomial \(-16 t^{2}+128\) gives the height of the pumpkin \(t\) seconds after it is dropped. Factor the greatest common factor from this binomial. (b) If the pumpkin is thrown down with initial velocity 32 feet per second, its height after \(t\) seconds is given by the trinomial \(-16 t^{2}-32 t+128\). Completely factor this trinomial.
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