type the correct answer in the box.\nwhat is the completely factored form of this polynomial?\n$18x^3…

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

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

Answer

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))