Question

factor out the gcf
\\24x^4 + 12x^3 - 36x\\

Explanation:

Identify the terms of the polynomial

The given polynomial expression is:

$$24x^4 + 12x^3 - 36x$$

The terms are \(24x^4\), \(12x^3\), and \(-36x\).

Find the GCF of the numerical coefficients

Determine the greatest common factor of the coefficients \(24\), \(12\), and \(36\):

• Factors of \(24\): \(1, 2, 3, 4, 6, 8, \mathbf{12}, 24\)
• Factors of \(12\): \(1, 2, 3, 4, 6, \mathbf{12}\)
• Factors of \(36\): \(1, 2, 3, 4, 6, 9, \mathbf{12}, 18, 36\)

The greatest common numerical factor is \(12\).

Find the GCF of the variable parts

Determine the greatest common factor of the variable parts \(x^4\), \(x^3\), and \(x\):

• The lowest power of \(x\) present in all terms is \(x^1\) (or \(x\)).

Thus, the greatest common variable factor is \(x\).

Combine to find the overall GCF

Multiply the numerical GCF and the variable GCF:

$$\text{GCF} = 12x$$

Factor out the GCF from each term

Divide each term of the original polynomial by \(12x\):

• \(\frac{24x^4}{12x} = 2x^3\)
• \(\frac{12x^3}{12x} = x^2\)
• \(\frac{-36x}{12x} = -3\)

Combine these remaining terms inside parentheses:

$$12x(2x^3 + x^2 - 3)$$

Answer:

$$12x(2x^3 + x^2 - 3)$$