Use the long division method to find the result when minus, 12, x, cubed, minus, 13, x, squared, minus, 6−12x 
3
 −13x 
2
 −6 is divided by 4, x, minus, 14x−1. If there is a remainder, express the result in the form q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fractionq(x)+ 
b(x)
r(x)
​
 .

We are asked to divide the polynomial \( -12x^3 - 13x^2 - 6 \) by \( 4x - 1 \) using the long division method.

The general division form can be expressed as:
\[
\frac{f(x)}{g(x)} = q(x) + \frac{r(x)}{g(x)}
\]
where \( f(x) \) is the dividend, \( g(x) \) is the divisor, \( q(x) \) is the quotient, and \( r(x) \) is the remainder.

Let’s go step by step:

### Step 1: Set up the division.

We want to divide \( -12x^3 - 13x^2 - 6 \) by \( 4x - 1 \). First, set up the long division:

\[
\frac{-12x^3 - 13x^2 - 6}{4x - 1}
\]

### Step 2: Divide the first term.

To begin, divide the leading term of the dividend \( -12x^3 \) by the leading term of the divisor \( 4x \). This gives:

\[
\frac{-12x^3}{4x} = -3x^2
\]

Now, write \( -3x^2 \) as the first term of the quotient.

### Step 3: Multiply and subtract.

Next, multiply \( -3x^2 \) by the divisor \( 4x - 1 \):

\[
(-3x^2)(4x - 1) = -12x^3 + 3x^2
\]

Now subtract this from the original polynomial:

\[
(-12x^3 - 13x^2 - 6) - (-12x^3 + 3x^2) = (-13x^2 - 6) - (3x^2) = -16x^2 - 6
\]

### Step 4: Divide the second term.

Now divide \( -16x^2 \) by \( 4x \):

\[
\frac{-16x^2}{4x} = -4x
\]

Write \( -4x \) as the next term of the quotient.

### Step 5: Multiply and subtract.

Next, multiply \( -4x \) by \( 4x - 1 \):

\[
(-4x)(4x - 1) = -16x^2 + 4x
\]

Now subtract this from the current polynomial:

\[
(-16x^2 - 6) - (-16x^2 + 4x) = (-6) - (4x) = -4x - 6
\]

### Step 6: Divide the third term.

Now divide \( -4x \) by \( 4x \):

\[
\frac{-4x}{4x} = -1
\]

Write \( -1 \) as the next term of the quotient.

### Step 7: Multiply and subtract.

Now, multiply \( -1 \) by \( 4x - 1 \):

\[
(-1)(4x - 1) = -4x + 1
\]

Subtract this from the current polynomial:

\[
(-4x - 6) - (-4x + 1) = -6 - 1 = -7
\]

### Step 8: Express the remainder.

Since \( -7 \) is the remainder, the final result of the division is:

\[
q(x) = -3x^2 - 4x - 1
\]
\[
r(x) = -7
\]

Thus, the complete result is:

\[
\frac{-12x^3 - 13x^2 - 6}{4x - 1} = -3x^2 - 4x - 1 + \frac{-7}{4x - 1}
\]

This is the solution.

Would you like more details on any of these steps?

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**Related Questions:**

1. What are the properties of polynomial division?
2. How can remainders affect the behavior of polynomial functions?
3. Can long division be applied to higher-degree polynomials effectively?
4. How is the quotient affected when dividing by a binomial?
5. Why is it important to arrange polynomials by descending powers when performing division?

**Tip:** Always ensure that the divisor and dividend are written in descending powers of \(x\) before starting the division process to avoid confusion.

Use the long division method to find the result when -12x^3 - 13x^2 - 6 is divided by 4x - 1. If there is a remainder, express the result in the form q(x) + r(x)/b(x).

Master the long division method to divide the polynomial -12x^3 - 13x^2 - 6 by 4x - 1. This detailed solution includes steps to find the quotient and remainder, covering key algebraic concepts suitable for Grades 10-12.

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