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Chapter 7: Problem 202

Factor. \(7 x^{2}-21 x\)

Short Answer

Expert verified
The factored form is \(7x(x - 3)\).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First, find the greatest common factor (GCF) of the terms in the polynomial. The terms are 7x虏 and -21x. The GCF of 7 and 21 is 7. Both terms also include the variable 'x', so the GCF is 7x.
02

Factor Out the GCF

Next, factor out the GCF (7x) from the polynomial. Divide each term by 7x: \(7x^2 \bigdiv 7x = x\) \(-21x \bigdiv 7x = -3\). Write the polynomial as a product of the GCF and the resulting polynomial inside parentheses: \(7x^2 - 21x = 7x(x - 3)\).
03

Verify the Factoring

Ensure the expression is correctly factored by distributing the GCF back into the factored polynomial: \(7x \times x = 7x^2\) \(7x \times -3 = -21x\). The original polynomial is recovered, confirming the factoring is correct: \(7x^2 - 21x = 7x(x - 3)\).

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Key Concepts

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

Greatest Common Factor (GCF)
The first step in factoring a polynomial is to find the Greatest Common Factor (GCF). The GCF is the highest number and variable combination that can evenly divide each term in the polynomial. For the polynomial given, which is \(7x^2 - 21x\), the GCF is found by identifying the highest common factors in the coefficients (7 and 21) and the variables \(x^2\) and \(x\).

Here鈥檚 how you can do it step-by-step:
  • Find the GCF of the numbers 7 and 21. Since the highest number that divides both 7 and 21 evenly is 7, it becomes part of the GCF.
  • For the variable part, you consider the lowest power of x present in each term. Here, both terms have the variable \(x\), so \(x\) is included in the GCF.
Thus, the GCF of 7 and 21, combined with \(x\), gives us a GCF of 7x.
Polynomial Division
After identifying the GCF, the next step is polynomial division. This involves dividing each term of the polynomial by the GCF.

Let鈥檚 apply this to the polynomial \(7x^2 - 21x\):
  • Divide \(7x^2\) by \(7x\): \ \ \(7x^2 \bigdiv 7x = x\).
  • Divide \(-21x\) by \(7x\): \ \ \(-21x \bigdiv 7x = -3\).
Write this as a product of the GCF and the resulting polynomial inside parentheses: \ \ \(7x^2 - 21x = 7x(x - 3)\).

In this process, you are effectively reversing the distribution of the GCF across the terms of the polynomial, simplifying it to a product of simpler terms.
Factoring Verification
The last step is verification, ensuring that your factored form is correct. This involves redistributing the GCF back into the polynomial to see if it matches the original expression. Here鈥檚 how you can check it:
  • Multiply \(7x\) with \(x\): \ \ \(7x \times x = 7x^2\).
  • Multiply \(7x\) with \(-3\): \ \ \(7x \times -3 = -21x\).
If your polynomial returns to its original form, in this case, \(7x^2 - 21x\), then your factorization is correct.

This verification process ensures that the distribution was done accurately and no terms were missed or incorrectly simplified.

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