Divide. Show your work. \frac{x^3 – 4x + 6}{x+3}
The Correct Answer and Explanation is:
To divide x3−4x+6x+3\frac{x^3 – 4x + 6}{x+3}x+3×3−4x+6, we will use polynomial long division. Here’s the step-by-step process:
Step 1: Set up the division
We want to divide x3−4x+6x^3 – 4x + 6×3−4x+6 by x+3x + 3x+3. Start by setting it up like a regular division problem:Dividend: x3+0x2−4x+6\text{Dividend: } x^3 + 0x^2 – 4x + 6Dividend: x3+0x2−4x+6Divisor: x+3\text{Divisor: } x + 3Divisor: x+3
Step 2: Divide the first term
Divide the first term of the dividend, x3x^3×3, by the first term of the divisor, xxx. This gives:x3x=x2\frac{x^3}{x} = x^2xx3=x2
Write x2x^2×2 as the first term of the quotient.
Step 3: Multiply and subtract
Now, multiply x2x^2×2 by the divisor x+3x + 3x+3 and subtract the result from the dividend:x2⋅(x+3)=x3+3x2x^2 \cdot (x + 3) = x^3 + 3x^2×2⋅(x+3)=x3+3×2
Subtract x3+3x2x^3 + 3x^2×3+3×2 from the dividend x3+0x2−4x+6x^3 + 0x^2 – 4x + 6×3+0x2−4x+6:(x3+0x2−4x+6)−(x3+3×2)=−3×2−4x+6(x^3 + 0x^2 – 4x + 6) – (x^3 + 3x^2) = -3x^2 – 4x + 6(x3+0x2−4x+6)−(x3+3×2)=−3×2−4x+6
Step 4: Divide the next term
Now, divide −3×2-3x^2−3×2 by xxx, which gives:−3x2x=−3x\frac{-3x^2}{x} = -3xx−3×2=−3x
Write −3x-3x−3x as the next term in the quotient.
Step 5: Multiply and subtract again
Multiply −3x-3x−3x by the divisor x+3x + 3x+3 and subtract:−3x⋅(x+3)=−3×2−9x-3x \cdot (x + 3) = -3x^2 – 9x−3x⋅(x+3)=−3×2−9x
Subtract −3×2−9x-3x^2 – 9x−3×2−9x from −3×2−4x+6-3x^2 – 4x + 6−3×2−4x+6:(−3×2−4x+6)−(−3×2−9x)=5x+6(-3x^2 – 4x + 6) – (-3x^2 – 9x) = 5x + 6(−3×2−4x+6)−(−3×2−9x)=5x+6
Step 6: Divide the next term
Next, divide 5x5x5x by xxx, which gives:5xx=5\frac{5x}{x} = 5x5x=5
Write 555 as the next term in the quotient.
Step 7: Multiply and subtract
Multiply 555 by x+3x + 3x+3 and subtract:5⋅(x+3)=5x+155 \cdot (x + 3) = 5x + 155⋅(x+3)=5x+15
Subtract 5x+155x + 155x+15 from 5x+65x + 65x+6:(5x+6)−(5x+15)=−9(5x + 6) – (5x + 15) = -9(5x+6)−(5x+15)=−9
Step 8: Write the final result
Now, we have no more terms left to divide by xxx. Therefore, the quotient is:x2−3x+5x^2 – 3x + 5×2−3x+5
And the remainder is −9-9−9. Thus, the final result is:x3−4x+6x+3=x2−3x+5+−9x+3\frac{x^3 – 4x + 6}{x + 3} = x^2 – 3x + 5 + \frac{-9}{x+3}x+3×3−4x+6=x2−3x+5+x+3−9
Final Answer:
x2−3x+5−9x+3\boxed{x^2 – 3x + 5 – \frac{9}{x+3}}x2−3x+5−x+39
This is the complete quotient with the remainder written as a fraction.
