Question

divide.
\\((6x^3 - 16x^2 - 4x - 15) \div (2x^2 - 6x)\\)
your answer should give the quotient and the remainder.
quotient:
remainder:

Explanation:

Step1: Divide leading terms

Divide the leading term of the dividend \(6x^3\) by the leading term of the divisor \(2x^2\). So, \(\frac{6x^3}{2x^2} = 3x\). This is the first term of the quotient.

Step2: Multiply and subtract

Multiply the divisor \(2x^2 - 6x\) by \(3x\): \(3x(2x^2 - 6x)=6x^3 - 18x^2\). Subtract this from the dividend: \((6x^3 - 16x^2 - 4x - 15)-(6x^3 - 18x^2)=2x^2 - 4x - 15\).

Step3: Divide leading terms again

Divide the leading term of the new dividend \(2x^2\) by the leading term of the divisor \(2x^2\). So, \(\frac{2x^2}{2x^2}=1\). This is the next term of the quotient.

Step4: Multiply and subtract again

Multiply the divisor \(2x^2 - 6x\) by \(1\): \(1(2x^2 - 6x)=2x^2 - 6x\). Subtract this from the new dividend: \((2x^2 - 4x - 15)-(2x^2 - 6x)=2x - 15\).
Since the degree of the remainder \(2x - 15\) is less than the degree of the divisor \(2x^2 - 6x\), we stop.

Answer:

Quotient: \(3x + 1\)
Remainder: \(2x - 15\)