Question 93·Hard·Equivalent Expressions

Try to express $x^{3}-4x^{2}+3x+6$ in the form $(x-2)\cdot(\text{some quadratic}) + \text{remainder}$, where the remainder is a constant.

Because the divisor is $x-2$, you can use synthetic division with $2$ and the coefficients $1, -4, 3, 6$ to quickly find the quotient and remainder.

Once you know the quotient polynomial and the remainder, rewrite the original fraction as that quotient plus the remainder over $x-2$, and look for the matching choice.

In Desmos, type f(x) = (x^3 - 4x^2 + 3x + 6) / (x - 2) to define the function given in the problem.

For each choice, define a new function equal to f(x) minus that choice, for example h_A(x) = f(x) - (x^2 - 2x - 1 - 4/(x-2)), h_B(x) = f(x) - ( ... ), and so on for all options.

Look at each h_* graph. If a choice is equivalent to the original expression, its difference function will lie exactly on the x-axis (i.e., be $y=0$) for all $x$ in the domain (excluding $x=2$ where there is a discontinuity). The option whose difference graph is identically zero is the correct one.

You are given

This is a polynomial divided by a linear binomial. In general, when you divide a polynomial (the dividend) by another polynomial (the divisor), you can rewrite it as

Then

So the goal is to find the quotient and remainder when dividing $x^{3}-4x^{2}+3x+6$ by $x-2$.

Because the divisor is $x-2$, you can use synthetic division with the number $2$.

1. Write the coefficients of the numerator in order: $1$ (for $x^3$), $-4$ (for $x^2$), $3$ (for $x$), and $6$ (constant).

2. Set up synthetic division with $2$ on the left:

   • Bring down the $1$.
   • Multiply $1$ by $2$ to get $2$, and add to $-4$ to get $-2$.
   • Multiply $-2$ by $2$ to get $-4$, and add to $3$ to get $-1$.
   • Multiply $-1$ by $2$ to get $-2$, and add to $6$ to get $4$.

So the synthetic division row becomes: $1, -2, -1, 4$.

In synthetic division, the numbers after you finish correspond to the coefficients of the quotient and then the remainder.

• The first three numbers $1, -2, -1$ are the coefficients of the quotient polynomial.
• Since the original numerator was degree 3, the quotient will be degree 2:
  • $1$ is the coefficient of $x^2$,
  • $-2$ is the coefficient of $x$,
  • $-1$ is the constant term.
• The last number, $4$, is the remainder.

So the quotient is a quadratic polynomial with coefficients $1, -2, -1$, and the remainder is $4$.

Using the division relationship

we plug in the specific quotient and remainder we found:

• Quotient: $x^2 - 2x - 1$
• Remainder: $4$
• Divisor: $x-2$

So

Comparing this with the answer choices, this matches $x^{2}-2x-1+\dfrac{4}{x-2}$, which is the correct option.