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Chapter 5: Problem 14

Write an equivalent expression by factoring out the greatest common factor. $$9 x^{2}-3 x+3$$

Short Answer

Expert verified
3(3x^{2} - x + 1)

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Look at each term in the expression to determine the greatest common factor. For the expression \( 9x^{2}-3x+3 \), the GCF of the numerical coefficients (9, -3, and 3) is 3.
02

Factor out the GCF

Divide each term in the expression by the GCF (3) and write the expression as the product of the GCF and the resulting polynomial. \[ 9x^{2}-3x+3 = 3(3x^{2} - x + 1) \]

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Key Concepts

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

greatest common factor
The concept of the greatest common factor (GCF) is central to various areas of mathematics, including simplifying algebraic expressions. It represents the largest factor that divides all the terms in a given expression equally. Identifying the GCF involves:
  • Looking at the coefficients of each term. For example, in the expression 9x^2 - 3x + 3, the coefficients are 9, -3, and 3.
  • Finding the GCF of these numbers. The factors of 9 are 1, 3, and 9. The factors of -3 are 1 and -3. The factors of 3 are 1 and 3.
  • Selecting the highest common factor from these sets. Here, the GCF is 3.
Once you identify the GCF, you can use it to simplify expressions, making them more understandable and manageable for further calculations.
factoring polynomials
Factoring polynomials involves rewriting an expression as a product of simpler polynomials. This technique is particularly useful when simplifying expressions and solving equations. For our example, the polynomial 9x^2 - 3x + 3 can be factored by:
  • Identifying the GCF, which is 3.
  • Dividing each term of the polynomial by this GCF to get the simpler polynomial. We divide 9x^2, -3x, and 3 by 3 to get 3x^2 - x + 1.
  • Rewriting the original polynomial as the product of the GCF and the simpler polynomial: 3(3x^2 - x + 1).
Mastering this method takes practice, but it helps immensely in solving more complex algebraic problems. It also lays the groundwork for techniques such as factoring quadratics and higher-degree polynomials.
algebraic expressions
Algebraic expressions are combinations of variables, numbers, and operations. They form the backbone of algebra and appear in almost every math problem. An algebraic expression like 9x^2 - 3x + 3 consists of:
  • Terms: Parts of the expression separated by plus or minus signs. Here, the terms are 9x^2, -3x, and 3.
  • Coefficients: Numerical factors in terms. The coefficients in this example are 9, -3, and 3.
  • Variables: Symbols representing unknown values. In this case, the variable is x.
Understanding how to manipulate these components is crucial. Factoring is one fundamental method. By factoring out the greatest common factor, we transform the given expression into a more convenient form. This simplification allows easier expansion, differentiation, or integration in advanced mathematics.

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