QuestionThe expression $\frac{6 x^{3}+17 x^{2}+10 x+2}{2 x+3}$ equals

Studdy Solution

STEP 1

What is this asking? We're asked to simplify a big fraction with polynomials by performing polynomial long division. Watch out! Keep track of your signs and don't rush the multiplication and subtraction steps!

STEP 2

1. Set up the long division.
2. Perform the long division.

STEP 3

Alright, let's set up our polynomial long division!
We put the numerator, $6x^3 + 17x^2 + 10x + 2$ , inside the division symbol, and the divisor, $2x + 3$ , outside.

STEP 4

First, we look at the leading terms: $6x^3$ and $2x$ .
We ask ourselves, "What do we multiply $2x$ by to get $6x^3$ ?".
The answer is $3x^2$ !
So, we write $3x^2$ above the division symbol.

STEP 5

Now, we multiply $3x^2$ by the entire divisor, $2x + 3$ , which gives us $3x^2 \cdot (2x + 3) = 6x^3 + 9x^2$ .
We write this below the numerator, lining up the terms with the same powers of $x$ .

STEP 6

Next, we subtract! $(6x^3 + 17x^2) - (6x^3 + 9x^2) = 8x^2$ .
Remember to subtract both terms!
We then bring down the next term from the numerator, which is $10x$ , to get $8x^2 + 10x$ .

STEP 7

We repeat the process!
We ask, "What do we multiply $2x$ by to get $8x^2$ ?".
The answer is $4x$ .
We write $4x$ above the division symbol, next to the $3x^2$ .

STEP 8

Multiply $4x$ by the divisor $2x + 3$ to get $4x \cdot (2x + 3) = 8x^2 + 12x$ .
We write this below $8x^2 + 10x$ .

STEP 9

Subtract again! $(8x^2 + 10x) - (8x^2 + 12x) = -2x$ .
Bring down the last term from the numerator, which is $2$ , to get $-2x + 2$ .

STEP 10

One last time!
What do we multiply $2x$ by to get $-2x$ ?
The answer is $-1$ .
Write $-1$ above the division symbol.

STEP 11

Multiply $-1$ by the divisor $2x + 3$ to get $-1 \cdot (2x + 3) = -2x - 3$ .
Write this below $-2x + 2$ .

STEP 12

Subtract! $(-2x + 2) - (-2x - 3) = 5$ .
This is our remainder.

STEP 13

Our quotient is $3x^2 + 4x - 1$ with a remainder of $5$ .
We write the final answer as $3x^2 + 4x - 1 + \frac{5}{2x + 3}$ .

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