When the polynomial 6x^3 - 7x^2 - 16x + 12 is factored completely into three linear factor… — ACT

When the polynomial $6x^3 - 7x^2 - 16x + 12$ is factored completely into three linear factors of the form $(ax + b)(cx + d)(ex + f)$ , where $a$ , $c$ , and $e$ are positive integers, what is the value of $a + b + c + d + e + f$ ?

Answer: 5

Answer

The value of the sum of the coefficients is 5.

The polynomial factors completely over the integers as

(x - 2)(2x + 3)(3x - 2)

. The sum of the six coefficients is

1 + (-2) + 2 + 3 + 3 + (-2) = 5

Step-by-Step Solution

Find one linear factor of the cubic polynomial using the Factor Theorem.

The root

x = 2

satisfies the equation, so

(x - 2)

is a factor.

Testing integer factors of the constant term 12 reveals that

x = 2

evaluates the polynomial to 0.

Perform synthetic division or polynomial long division to divide the cubic by the linear factor.

The quotient is the quadratic expression

6x^2 + 5x - 6

This reduces the degree of the polynomial to allow quadratic factoring techniques.

Factor the quadratic quotient into two linear binomials.

The quadratic factors into

(2x + 3)(3x - 2)

Using the AC method,

6 \times (-6) = -36

, and the factors of

-36

that sum to

5

are

9

and

-4

Identify the coefficients and sum them.

The sum is

1 + (-2) + 2 + 3 + 3 + (-2) = 5

The factors are

(1x - 2)(2x + 3)(3x - 2)

, corresponding to the coefficients

a=1, b=-2, c=2, d=3, e=3, f=-2

Key Concept

Complete factorization of cubic polynomials with integer coefficients using the Rational Root Theorem and quadratic factoring.

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