Question

divide.
\\((10x^2 + 23x + 12) \div (2x + 3)\\)
your answer should give the quotient and the remainder.
quotient: \\(square\\)
remainder: \\(square\\)

Explanation:

Step1: Use polynomial long division

Divide the leading term of the dividend \(10x^2 + 23x + 12\) by the leading term of the divisor \(2x + 3\). The leading term of the dividend is \(10x^2\) and the leading term of the divisor is \(2x\). So, \(\frac{10x^2}{2x}=5x\). This is the first term of the quotient.

Step2: Multiply the divisor by \(5x\)

Multiply \(2x + 3\) by \(5x\): \(5x(2x + 3)=10x^2+15x\).

Step3: Subtract this from the dividend

Subtract \(10x^2 + 15x\) from \(10x^2 + 23x + 12\): \((10x^2 + 23x + 12)-(10x^2 + 15x)=8x + 12\).

Step4: Divide the new leading term by the divisor's leading term

Now, divide the leading term of \(8x + 12\) (which is \(8x\)) by the leading term of \(2x + 3\) (which is \(2x\)). So, \(\frac{8x}{2x}=4\). This is the next term of the quotient.

Step5: Multiply the divisor by \(4\)

Multiply \(2x + 3\) by \(4\): \(4(2x + 3)=8x + 12\).

Step6: Subtract this from the new dividend

Subtract \(8x + 12\) from \(8x + 12\): \((8x + 12)-(8x + 12)=0\). Since the remainder is \(0\), we stop here.

Answer:

Quotient: \(5x + 4\)
Remainder: \(0\)