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Chapter 7: Problem 27

Factor the greatest common factor from each polynomial. \(9 n-63\)

Short Answer

Expert verified
The factored form is \(9(n - 7)\).

Step by step solution

01

Identify the greatest common factor (GCF)

Identify the greatest common factor of the terms in the polynomial. The terms are 9 and -63. The factors of 9 are 1, 3, 9, and the factors of -63 are 1, 3, 7, 9, 21, 63. The greatest common factor is 9.
02

Factor out the GCF

Factor 9 out of each term in the polynomial. This gives us: \[9n - 63 = 9(n - 7)\]

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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 when working with polynomials. The GCF is the largest number that divides exactly into each term of the polynomial without leaving a remainder.
If we look at the polynomial in the exercise, consisting of the terms 9n and -63, here's how we find the GCF:
  • List the factors of 9: 1, 3, 9
  • List the factors of -63: 1, 3, 7, 9, 21, 63

The common factors are 1, 3, and 9, making the greatest common factor 9.
Once we have identified the GCF, we proceed to factor it out from the polynomial.
Polynomials
Polynomials are expressions consisting of variables and coefficients, connected through addition, subtraction, or multiplication. Each part of the polynomial is known as a term.
In our exercise, the polynomial is 9n - 63, where 9n and -63 are the terms.
When working with polynomials, it is helpful to understand the properties of terms and operations involving them.
  • Terms are separated by addition or subtraction signs
  • Like terms have the same variables raised to the same power
  • Operations on polynomials often involve combining like terms or factoring
Factoring Steps
Factoring polynomials involves writing the polynomial as a product of its factors. These steps can help you understand how to factor a polynomial:
  • Step 1: Identify the greatest common factor (GCF)
  • Step 2: Rewrite the polynomial as a product of the GCF and the remaining polynomial

Using our exercise as an example, we have the polynomial 9n - 63, with the GCF identified as 9.
The next step is to factor out the GCF. This means:
\(9n - 63 = 9(n - 7)\)
Here, we simply divided each term by the GCF 9 and wrote the polynomial as a product: 9 times (n - 7).
This factoring helps to simplify and solve the polynomial more easily.

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