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Chapter 10: Problem 619

Factor the greatest common factor from each polynomial. $$15 r+35$$

Short Answer

Expert verified
5(3r + 7)

Step by step solution

01

Identify the coefficients and terms

Look at the polynomial \[15r + 35\].You have two terms: 15r and 35. Identify the coefficients: 15 and 35.
02

Find the Greatest Common Factor (GCF)

Determine the GCF of the coefficients 15 and 35. - The factors of 15: 1, 3, 5, 15.- The factors of 35: 1, 5, 7, 35.The greatest common factor is 5.
03

Factor out the GCF

Divide each term by the GCF and write the polynomial as a product of the GCF and a binomial: \[15r \text{ divided by } 5 = 3r,\]\[35 \text{ divided by } 5 = 7.\]Factor the GCF out: \[15r + 35 = 5(3r + 7)\]

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Key Concepts

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

Greatest Common Factor
In algebra, the greatest common factor (GCF) is an important tool to simplify polynomials. The GCF of two or more numbers is the largest number that divides each of those numbers without leaving a remainder.

For example, in the polynomial \(15r + 35\), we look at the coefficients 15 and 35. To find their GCF:
  • Identify the factors of each number.
  • The factors of 15 are: 1, 3, 5, 15.
  • The factors of 35 are: 1, 5, 7, 35.
By comparing these factors, we see that the largest common factor is 5. This is our GCF, which we use to factor the polynomial.
Coefficients
Coefficients are the numerical factors in terms of polynomials. They multiply the variable part of each term.

In the polynomial \(15r + 35\), the coefficients are 15 and 35.

Here's how to identify and use coefficients:
  • First, pick out the coefficients from each term. In our polynomial, look at 15 in \(15r\) and 35 in the constant term.
  • Next, we use these coefficients to find the greatest common factor (GCF), as explained in the previous section.
  • Once the GCF is identified, it can simplify the polynomial by extracting it.
Identifying coefficients helps us understand the structure of the polynomial and apply operations like factoring.
Binomial
A binomial is a polynomial with two terms, typically separated by a plus or minus sign. In factored form, a polynomial often turns into a binomial.

For example, our polynomial \(15r + 35\) is simplified by factoring out the GCF, 5.

After dividing each term by the GCF, we get:
  • \(15r \div 5 = 3r\)
  • \(35 \div 5 = 7\)
This simplifies the polynomial to: \(5(3r + 7)\). The expression \(3r + 7\) is a binomial because it has exactly two terms, 3r and 7. Understanding binomials is crucial in algebra as they are the basic building blocks of many more complex expressions and equations.

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