Problem 25 - Introductory Algebra for College Students

Question

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state. $$14 x^{3}+21 x^{2}$$

Step-by-Step Solution

Answer

The factored form of \(14x^3 + 21x^2 \) using the GCF is \(7x^2(2x + 3)\).

Step 1: Identify the terms in the polynomial

The polynomial provided in the question is \( 14x^3 + 21x^2 \). This polynomial has two terms - \(14x^3\) and \(21x^2\).

Step 2: Find the greatest common factor (GCF)

For \(14x^3\) and \(21x^2\), the GCF is the largest number that divides both 14 and 21 and the smallest power of x common to both terms. Looking at the coefficients, 7 is the largest number that divides both 14 and 21. From the powers of x, \(x^2\) is common to both terms. Therefore, the GCF is \(7x^2\).

Step 3: Divide each term by the GCF

Divide each term in the polynomial by the GCF to find the remaining factor. So, \(14x^3 ÷ 7x^2 = 2x \) and \(21x^2 ÷ 7x^2 = 3\).

Step 4: Write the polynomial as a product of the GCF and the remaining factor

The polynomial \(14x^3 + 21 x^2 \) can now be written as the GCF (\(7x^2\)) times the remaining factor (\(2x + 3\)). Therefore, \(14x^3 + 21 x^2 = 7x^2(2x + 3)\).