Question

type the correct answer in the box.
what is the completely factored form of this polynomial?
$18x^3 - 120x^2 - 42x$
write the greatest common factor (gcf) first, and write all factors in standard form.

Explanation:

Step1: Find the GCF of coefficients and variables

For coefficients 18, -120, -42: GCF of 18, 120, 42 is 6 (since \(18 = 6\times3\), \(120 = 6\times20\), \(42 = 6\times7\)). For variable \(x\) terms: \(x^3\), \(x^2\), \(x\) have GCF \(x\). So overall GCF is \(6x\).

Step2: Factor out GCF from polynomial

Divide each term by \(6x\): \(\frac{18x^3}{6x}=3x^2\), \(\frac{-120x^2}{6x}=-20x\), \(\frac{-42x}{6x}=-7\). So the polynomial becomes \(6x(3x^2 - 20x - 7)\).

Step3: Factor the quadratic

Factor \(3x^2 - 20x - 7\). Find two numbers \(a,b\) such that \(a\times b = 3\times(-7)= -21\) and \(a + b = -20\). The numbers are -21 and 1. Rewrite middle term: \(3x^2 -21x + x -7\). Group: \((3x^2 -21x)+(x -7)=3x(x -7)+1(x -7)=(3x + 1)(x -7)\).

Step4: Combine all factors

Putting it together, the completely factored form is \(6x(3x + 1)(x - 7)\).

Answer:

\(6x(3x + 1)(x - 7)\)