Anja Divides \left(8x^3 - 36x^2 + 54x - 27\right ] By ( 2 X − 3 (2x - 3 ( 2 X − 3 ] As Shown Below. What Error Does Anja Make?$ \begin{array}{r} 2x - 3 \ \ \ \ \ \ \longdiv { 8x^3 - 36x^2 + 54x - 27 } \ \phantom{2x - 3 \ \ } \ \ \ \ \ \ 8x^3 -

Introduction

Polynomial division is a fundamental concept in algebra, used to divide a polynomial by another polynomial. It is an essential tool for simplifying complex expressions and solving equations. However, like any mathematical operation, it requires careful attention to detail to avoid errors. In this article, we will examine a polynomial division problem and identify the mistake made by Anja.

The Problem

Anja is attempting to divide the polynomial $\left(8x^3 - 36x^2 + 54x - 27\right)$ by $(2x - 3)$. The division is shown below:

\begin{array}{r} 2x - 3 \ \ \ \ \ \ \longdiv { 8x^3 - 36x^2 + 54x - 27 } \ \phantom{2x - 3 \ \ } \ \ \ \ \ \ 8x^3 - \end{array}

The Error

The first step in polynomial division is to divide the leading term of the dividend by the leading term of the divisor. In this case, the leading term of the dividend is $8x^3$ and the leading term of the divisor is $2x$. Therefore, the first term of the quotient should be $\frac{8x^3}{2x} = 4x^2$.

However, Anja's work shows that she has written $8x^3 - 36x^2$ as the result of the first step. This is incorrect because she has not multiplied the divisor $(2x - 3)$ by the quotient term $4x^2$.

Correcting the Error

To correct the error, we need to multiply the divisor $(2x - 3)$ by the quotient term $4x^2$. This gives us:

$(2x - 3) \cdot 4x^2 = 8x^3 - 12x^2$

Now, we can subtract this result from the dividend:

$8x^3 - 36x^2 + 54x - 27 - (8x^3 - 12x^2) = -24x^2 + 54x - 27$

The next step in polynomial division is to divide the leading term of the result by the leading term of the divisor. In this case, the leading term of the result is $-24x^2$ and the leading term of the divisor is $2x$. Therefore, the next term of the quotient should be $\frac{-24x^2}{2x} = -12x$.

The Correct Quotient

Using the corrected result from the previous step, we can continue the polynomial division:

Q: What is the mistake that Anja made in the polynomial division problem?

A: The mistake that Anja made is that she did not multiply the divisor $(2x - 3)$ by the quotient term $4x^2$ in the first step of the division.

Q: How do you correct the error in the polynomial division problem?

A: To correct the error, we need to multiply the divisor $(2x - 3)$ by the quotient term $4x^2$. This gives us:

$(2x - 3) \cdot 4x^2 = 8x^3 - 12x^2$

Then, we can subtract this result from the dividend:

$8x^3 - 36x^2 + 54x - 27 - (8x^3 - 12x^2) = -24x^2 + 54x - 27$

Q: What is the next step in the polynomial division problem?

A: The next step in the polynomial division problem is to divide the leading term of the result by the leading term of the divisor. In this case, the leading term of the result is $-24x^2$ and the leading term of the divisor is $2x$. Therefore, the next term of the quotient should be $\frac{-24x^2}{2x} = -12x$.

Q: How do you continue the polynomial division problem?

A: Using the corrected result from the previous step, we can continue the polynomial division:

\begin{array}{r} 2x - 3 \ \ \ \ \ \ \longdiv { -24x^2 + 54x - 27 } \ \phantom{2x - 3 \ \ } \ \ \ \ \ \ -24x^2 + 12x \ \phantom{2x - 3 \ \ } \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \