Question

divide.
\\((2x^2 + 17x + 32) \div (x + 6)\\)
your answer should give the quotient and the remainder.
quotient: \\(\square\\)
remainder: \\(\square\\)

Explanation:

Step1: Divide the leading terms

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

Step2: Multiply and subtract

Multiply the divisor \((x + 6)\) by \(2x\) to get \(2x(x + 6)=2x^2+12x\). Subtract this from the dividend: \((2x^2 + 17x + 32)-(2x^2+12x)=5x + 32\).

Step3: Divide the new leading term

Divide the leading term of \(5x + 32\) (which is \(5x\)) by the leading term of the divisor \(x\). So, \(\frac{5x}{x}=5\). This is the next term of the quotient.

Step4: Multiply and subtract again

Multiply the divisor \((x + 6)\) by \(5\) to get \(5(x + 6)=5x+30\). Subtract this from \(5x + 32\): \((5x + 32)-(5x + 30)=2\). This is the remainder.

Answer:

Quotient: \(2x + 5\)
Remainder: \(2\)