What is the simpler form of the following expression?
(–18x3 + 17x + 6) ÷ (3x + 2)
A. –6x2 + 4x + 3
B. 6x2 + 4x – 3
C. –6x2 + 4x – 3
D. 6x2 – 4x + 3
C. –6x2 + 4x – 3
To find the simpler form of the expression (–18x^3 + 17x + 6) ÷ (3x + 2), we can use polynomial long division. Here are the steps to follow:
Step 1: Divide the first term in the numerator (-18x^3) by the first term in the denominator (3x). This gives us -6x^2.
Step 2: Multiply the entire denominator (3x + 2) by -6x^2. This gives us -18x^3 - 12x^2.
Step 3: Subtract this product (-18x^3 - 12x^2) from the original numerator (-18x^3 + 17x + 6). This gives us 12x^2 + 17x + 6.
Step 4: Repeat steps 1 to 3 using the simplified numerator (12x^2 + 17x + 6) as the new numerator.
Step 1: Divide the first term in the new numerator (12x^2) by the first term in the denominator (3x). This gives us 4x.
Step 2: Multiply the entire denominator (3x + 2) by 4x. This gives us 12x^2 + 8x.
Step 3: Subtract this product (12x^2 + 8x) from the new numerator (12x^2 + 17x + 6). This gives us 9x + 6.
At this point, we can see that there is no remainder, so our simplified form is: -6x^2 + 4x + 3.
Therefore, the answer is A. -6x^2 + 4x + 3.