Problem 474 Factor the greatest common facto... [FREE SOLUTION]

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Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 474

Factor the greatest common factor from each polynomial. $$4 q^{2}+24 q+28$$

Short Answer

Expert verified

4(q^2 + 6q + 7)

Step by step solution

Identify the Greatest Common Factor (GCF)

Examine each term in the polynomial \[4q^2, 24q, \text{ and } 28\] The GCF is the greatest factor that each of these terms share. Look at the coefficients first (4, 24, 28) and find that the GCF of these numbers is 4. There is no variable that is common to all three terms, so the GCF is 4.

Factor out the GCF

Divide each term of the polynomial by the GCF (4) and write the polynomial as \[4(q^2 + 6q + 7)\]. The polynomial inside the parentheses results from the division: \[\frac{4q^2}{4} = q^2\], \[\frac{24q}{4} = 6q\], and \[\frac{28}{4} = 7\].

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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 greatest common factor, or GCF, is the largest number that can evenly divide all terms in a polynomial. To find the GCF:

Identify the coefficients in the polynomial.In this example, the coefficients are 4, 24, and 28.
Determine the highest number that can divide each coefficient evenly. Here, the highest number is 4.

There aren鈥檛 any variables (like q) that are common to all terms. So, in $4q^2 + 24q + 28$, the GCF is simply 4, which lets us factor more easily.

Polynomial Division

Polynomial division is used to simplify a polynomial by removing its GCF. For the polynomial $4q^2 + 24q + 28$:

Divide each term by the GCF, which we've found is 4.
This means you divide 4q^2 by 4, 24q by 4, and 28 by 4.
Mathematically, \frac{4q^2}{4} = q^2 \, \frac{24q}{4} = 6q \, and \frac{28}{4} = 7 \.

After these divisions, you'll get the simplified polynomial $q^2 + 6q + 7$. This step helps break down the polynomial, making it easier to factor out completely.

Factoring Process

The factoring process involves simplifying a polynomial by removing common factors. For the polynomial $4q^2 + 24q + 28$, follow these steps:

First, identify the GCF, as we found out, it's 4.
Once identified, divide each term by this GCF.
Factor out the GCF and rewrite the polynomial. Thus, the polynomial becomes 4(q^2 + 6q + 7).

After factoring out, $4(q^2 + 6q + 7)$ is the complete factored form. The polynomial within parentheses is now easier to manage for any further factoring or solving techniques.

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91影视

Factor the greatest common factor from each polynomial. $$4 q^{2}+24 q+28$$

Short Answer

Step by step solution

Identify the Greatest Common Factor (GCF)

Factor out the GCF

Key Concepts

Greatest Common Factor (GCF)

Polynomial Division

Factoring Process

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