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Chapter 0: Problem 123

Using an example, explain how to factor out the greatest common factor of a polynomial.

Short Answer

Expert verified
The factored form of the polynomial \(12x^3 - 36x^2 + 24x\) by factoring out the GCF is \(12x^2(x - 3 + 2). \)

Step by step solution

01

Determine the factors

Let's take the polynomial \(12x^3 - 36x^2 + 24x\). Start by listing the factors of each coefficient i.e. For 12, the factors are 1, 2, 3, 4, 6, and 12. For 36, the factors are 1, 2, 3, 4, 6, 9, 12, 18, and 36. For 24, the factors are 1, 2, 3, 4, 6, 8, 12, and 24. Also list the factors of the variable part.
02

Find the greatest common factor (GCF)

For the coefficients, the largest factor common to 12, 36, and 24 is 12. In terms of variables, the degree of the polynomial is determined by the term with the highest power of x, which in this case is 3. Hence, 12 and \(x^2\) are the greatest common factors.
03

Factor the polynomial

To factorize the polynomial, each term needs to be divided by the GCF. So the polynomial \(12x^3 - 36x^2 + 24x\) becomes \(12x^2(x - 3 + 2)\) when factored out by the GCF.

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Key Concepts

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

Greatest Common Factor
The greatest common factor (GCF) is a fundamental concept in factoring polynomials. It refers to the largest number or expression that divides each term within a polynomial without a remainder. To identify the GCF, we first determine the factors of each term in the polynomial. Consider the polynomial example from the exercise: \(12x^3 - 36x^2 + 24x\).
  • For the coefficients 12, 36, and 24, we list all possible factors:
    • 12: 1, 2, 3, 4, 6, 12
    • 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
    • 24: 1, 2, 3, 4, 6, 8, 12, 24
  • The greatest common factor among these coefficients is 12.
  • For the variable part, each term contains at least \(x\), but we take the lowest power from all terms, which is \(x^1\).
Combining these, the greatest common factor of the polynomial is \(12x\). Understanding how to find the GCF is essential because it simplifies the process of breaking down complex expressions into more manageable pieces.
Polynomial Coefficients
Polynomial coefficients are the numerical factors in terms of a polynomial. In the polynomial \(12x^3 - 36x^2 + 24x\), the coefficients are 12, -36, and 24. These numbers play a crucial role as they determine the strength and weight of each term. Recognizing the coefficients is the first step in locating the greatest common factor (GCF). When we list out the coefficients' factors, we uncover common trends that help us simplify the entire polynomial.
  • It's essential to remember that coefficients can be positive or negative, so always take absolute values when finding the GCF initially, before considering the term signs in the final expression.
Working with coefficients demands precision; any mistake here affects the entire solution. Factoring polynomials starts strong and builds up from these numbers, making it an important foundational skill in algebra.
Factoring Process
The factoring process involves breaking down a polynomial into simpler terms that, when multiplied, reproduce the original expression. Once you've determined the greatest common factor (GCF), use it to factor the polynomial. Taking \(12x^3 - 36x^2 + 24x\) as an example:
  • The GCF is \(12x\).
  • We divide each term by the GCF:
    • \(12x^3 梅 12x = x^2\)
    • \(-36x^2 梅 12x = -3x\)
    • \(24x 梅 12x = 2\)
  • This yields the simplified polynomial: \(12x(x^2 - 3x + 2)\).
Factoring helps in simplifying expressions and solving polynomial equations effectively. It involves careful division and rearrangement of terms, stressing the importance of understanding mathematical operations over mere computation. As you practice, the factoring process becomes a powerful tool for handling polynomial complexities.

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Most popular questions from this chapter

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. You are choosing between two texting plans. Plan A has a monthly fee of \(\$ 15\) with a charge of \(\$ 0.08\) per text. Plan \(\mathbf{B}\) has a monthly fee of \(\$ 3\) with a charge of \(\$ 0.12\) per text. How many text messages in a month make plan A the better deal?

Explain how to factor the difference of two squares. Provide an example with your explanation.

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