https://file.aiquickdraw.com/mathbot/file/17263432958813hwin15j.jpg

We are given the equation:

\[
12x^3 - 31x^2 + 2x + 24 = (x - 2)(ax + b)(cx + d)
\]

where \(a\), \(b\), \(c\), and \(d\) are integers, and \(a > c\). We need to find the values of \(a\), \(b\), \(c\), and \(d\), and fully justify the answer.

### Step 1: Expand the right-hand side

We will first expand the right-hand side of the equation:

\[
(x - 2)(ax + b)(cx + d)
\]

Start by expanding \((ax + b)(cx + d)\):

\[
(ax + b)(cx + d) = acx^2 + adx + bcx + bd = acx^2 + (ad + bc)x + bd
\]

Now, expand \((x - 2)(acx^2 + (ad + bc)x + bd)\):

\[
(x - 2)(acx^2 + (ad + bc)x + bd) = x(acx^2 + (ad + bc)x + bd) - 2(acx^2 + (ad + bc)x + bd)
\]

First, distribute \(x\) to all terms inside the parentheses:

\[
x(acx^2 + (ad + bc)x + bd) = acx^3 + (ad + bc)x^2 + bdx
\]

Now, distribute \(-2\) to all terms inside the parentheses:

\[
-2(acx^2 + (ad + bc)x + bd) = -2acx^2 - 2(ad + bc)x - 2bd
\]

### Step 2: Combine like terms

Now, add the two expansions together:

\[
acx^3 + (ad + bc)x^2 + bdx - 2acx^2 - 2(ad + bc)x - 2bd
\]

Simplify:

\[
acx^3 + (ad + bc - 2ac)x^2 + (bd - 2(ad + bc))x - 2bd
\]

### Step 3: Compare coefficients

Now compare the terms of this expanded expression with the left-hand side \(12x^3 - 31x^2 + 2x + 24\). By comparing the coefficients of like powers of \(x\), we get the following system of equations:

1. Coefficient of \(x^3\): \(ac = 12\)
2. Coefficient of \(x^2\): \(ad + bc - 2ac = -31\)
3. Coefficient of \(x\): \(bd - 2(ad + bc) = 2\)
4. Constant term: \(-2bd = 24\)

### Step 4: Solve the system of equations

#### From equation (1):
\[
ac = 12
\]

Possible factor pairs for 12 are:
\[
(1, 12), (2, 6), (3, 4), (-1, -12), (-2, -6), (-3, -4)
\]

Since we are given that \(a > c\), we try \(a = 6\) and \(c = 2\).

#### From equation (4):
\[
-2bd = 24 \implies bd = -12
\]

Possible factor pairs for \(-12\) are:
\[
(1, -12), (-1, 12), (2, -6), (-2, 6), (3, -4), (-3, 4)
\]

#### From equation (2):
\[
ad + bc - 2ac = -31
\]

Substitute \(a = 6\), \(c = 2\), and \(ac = 12\):

\[
6d + 2b - 24 = -31 \implies 6d + 2b = -7
\]

This simplifies to:
\[
3d + b = -7 \quad \text{(Equation A)}
\]

#### Now solve using the factor pairs for \(bd = -12\):

- If \(b = -4\) and \(d = 3\):
    - Check if they satisfy Equation A:
    \[
    3(3) + (-4) = 9 - 4 = 5 \quad \text{(Does not work)}
    \]

- If \(b = -6\) and \(d = 2\):
    - Check if they satisfy Equation A:
    \[
    3(2) + (-6) = 6 - 6 = 0 \quad \text{(Does not work)}
    \]



It is given that 12x^3 - 31x^2 + 2x + 24 = (x - 2)(ax + b)(cx + d), where a, b, c and d are integers, and a > c. Find the values of a, b, c and d. Fully justify your answer.

This problem involves factoring a cubic polynomial expression and solving for unknowns a, b, c, and d by comparing coefficients. Learn how to approach such problems using algebraic expansion and solving systems of equations. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation