Question

(54x² + 45x + 8) ÷ (3x + 2)
answer should give the quotient and the
quotient:
remainder:

Explanation:

Step1: Divide the leading terms

Divide the leading term of the dividend \(54x^2\) by the leading term of the divisor \(3x\). So, \(\frac{54x^2}{3x} = 18x\). This is the first term of the quotient.

Step2: Multiply and subtract

Multiply the divisor \((3x + 2)\) by \(18x\): \(18x(3x + 2)=54x^2 + 36x\). Subtract this from the dividend: \((54x^2 + 45x + 8)-(54x^2 + 36x)=9x + 8\).

Step3: Divide the new leading term

Now, divide the leading term of \(9x + 8\) (which is \(9x\)) by the leading term of the divisor \(3x\). So, \(\frac{9x}{3x}=3\). This is the next term of the quotient.

Step4: Multiply and subtract again

Multiply the divisor \((3x + 2)\) by \(3\): \(3(3x + 2)=9x + 6\). Subtract this from \(9x + 8\): \((9x + 8)-(9x + 6)=2\). The remainder is \(2\) since the degree of the remainder (\(0\)) is less than the degree of the divisor (\(1\)).

Answer:

Quotient: \(18x + 3\)
Remainder: \(2\)