Question

7.4 factoring polynomials using greatest common factors quiz

factor: (21x^3 - 14x^2 + 7x)

( ) (7x(3x^2 - 2x + 1))
( ) (7x(3x - 2))
( ) (7x(3x^2 + 2x - 1))
( ) (x(21x^2 - 14x + 1))

Explanation:

Identify the terms of the polynomial

We begin with the given polynomial expression:

$$21x^3 - 14x^2 + 7x$$

The expression consists of three distinct terms:

1. First term: \(21x^3\)
2. Second term: \(-14x^2\)
3. Third term: \(7x\)

Find the GCF of the coefficients

We find the greatest common factor (GCF) of the numerical coefficients \(21\), \(14\), and \(7\):

• Factors of \(21\): \(1, 3, 7, 21\)
• Factors of \(14\): \(1, 2, 7, 14\)
• Factors of \(7\): \(1, 7\)

The greatest common numerical factor is \(7\).

Find the GCF of the variable parts

Next, we look at the variable parts of each term: \(x^3\), \(x^2\), and \(x\).
The lowest power of \(x\) present in all terms is \(x^1\) (or simply \(x\)).
Therefore, the greatest common variable factor is \(x\).

Determine the overall GCF

Combining the numerical and variable parts, the overall greatest common factor (GCF) for the polynomial is:

$$\text{GCF} = 7x$$

Divide each term by the GCF

Now, we divide each term of the original polynomial by \(7x\) to find the remaining polynomial factor:

• First term: \(\frac{21x^3}{7x} = 3x^2\)
• Second term: \(\frac{-14x^2}{7x} = -2x\)
• Third term: \(\frac{7x}{7x} = 1\)

Combining these results inside the parentheses gives:

$$7x(3x^2 - 2x + 1)$$

Answer:

• (A) \(7x(3x^2-2x+1)\) (Correct answer)
• (B) \(7x(3x-2)\)
• (C) \(7x(3x^2+2x-1)\)
• (D) \(x(21x^2-14x+1)\)