Question

divide using polynomial long division.
(7x³ + x² + x) ÷ (x² + 1) = □

Explanation:

Step1: Divide the leading terms

Divide the leading term of the dividend \(7x^3\) by the leading term of the divisor \(x^2\). So, \(\frac{7x^3}{x^2} = 7x\). This is the first term of the quotient.

Step2: Multiply and subtract

Multiply the divisor \(x^2 + 1\) by \(7x\): \(7x(x^2 + 1)=7x^3 + 7x\). Now, subtract this from the dividend: \((7x^3 + x^2 + x)-(7x^3 + 7x)=x^2 - 6x\).

Step3: Divide the new leading terms

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

Step4: Multiply and subtract again

Multiply the divisor \(x^2 + 1\) by \(1\): \(1(x^2 + 1)=x^2 + 1\). Subtract this from the new dividend: \((x^2 - 6x)-(x^2 + 1)=-6x - 1\).

Step5: Write the final result

The quotient is \(7x + 1\) and the remainder is \(-6x - 1\). So, \(\frac{7x^3 + x^2 + x}{x^2 + 1}=7x + 1+\frac{-6x - 1}{x^2 + 1}\).

Answer:

\(7x + 1+\frac{-6x - 1}{x^2 + 1}\) (or \(7x + 1-\frac{6x + 1}{x^2 + 1}\))