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Chapter 6: Problem 54

Write an equivalent expression by factoring out the greatest common factor. $$ 15 x^{2}-5 x+5 $$

Short Answer

Expert verified
The factored expression is \ 5(3x^{2} - x + 1) \.

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Determine the GCF of the coefficients and the variable terms. The coefficients are 15, -5, and 5. The GCF of these numbers is 5. The variable terms include x, so consider the lowest power of x, which is x^0 (no x, so no variable common factor). The GCF here is 5.
02

Factor Out the GCF

Factor 5 out from each term in the polynomial. This requires dividing each term by 5:\[ 15x^{2} \rightarrow 5 \times 3x^{2} \ -5x \rightarrow 5 \times (-x) \ 5 \rightarrow 5 \times 1 \ \text{So the polynomial becomes} \ 5(3x^{2} - x + 1) \]

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Key Concepts

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

greatest common factor
Factoring polynomials starts with identifying the Greatest Common Factor (GCF). The GCF is the largest factor that divides all terms in the polynomial.

To find the GCF, consider both the numerical coefficients and the variable parts. In the example **15x虏 - 5x + 5**, we look at each coefficient: 15, -5, and 5. The GCF of these numbers is 5.

For the variable terms, since the lowest power of x present is x鈦� (which means no 'x'), there is no common factor for the variable. Therefore, here, the GCF is simply 5. By identifying the GCF correctly, we make the next steps of factoring smoother and more structured.
polynomial expressions
Understanding polynomial expressions is key to mastering factoring. A polynomial is a sum of terms, each consisting of a variable raised to a non-negative integer power multiplied by a coefficient.

For example, in the expression **15x虏 - 5x + 5**, each term follows this structure:
  • 15x虏: 15 is the coefficient, x is the variable, and 虏 is the power.
  • -5x: -5 is the coefficient, x is the variable, and 1 is the power.
  • 5: 5 is the coefficient, with no variable, implying the power of x is 0.
Understanding this structure is essential for both identifying common factors and factoring polynomials effectively.
factoring
Factoring involves rewriting a polynomial as a product of simpler polynomials or factors. Once you have identified the greatest common factor, you can factor it out. Here鈥檚 how you factor the initial polynomial:

Take the polynomial **15x虏 - 5x + 5**. We determined the GCF is 5. Now, divide each term by this GCF:
  • 15x虏 divided by 5 gives 3x虏
  • -5x divided by 5 gives -x
  • 5 divided by 5 gives 1
So, the factored form is 5(3x虏 - x + 1).

By factoring, we simplify the polynomial and can solve or analyze it more easily. Factoring is foundational in algebra, helping solve equations and simplify expressions.

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Most popular questions from this chapter

Ashlee factors \(8 x^{2} y-10 x y^{2}\) as $$ 2 x y \cdot 4 x-2 x y \cdot 5 y $$ Is this the factorization of the polynomial? Why or why not?

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 36-(x+y)^{2} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ y^{2}-100 $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ x^{2}+2 x y+y^{2}-9 $$

Write an equivalent expression by factoring. Assume that all exponents are natural numbers. $$ 3 a^{n+1}+6 a^{n}-15 a^{n+2} $$
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