Q. \( \frac{x^3 + x^2 + x + 2}{x^2 – 1} \) long division

Q: Can I use synthetic division here?

Not the standard synthetic division: it works for linear divisors x - c. For quadratic divisor use polynomial long division or generalized synthetic method.

Q: Can the remainder be simplified using factorization of the denominator?

Yes: x^2-1 = (x-1)(x+1). Decompose (2x+3)/(x^2-1) = A/(x-1) + B/(x+1) with = 5/2, B = −1/2.

Q: What are the vertical and oblique (slant) asymptotes?

Vertical asymptotes at x = 1 and x = −1 (no cancellations). Oblique asymptote is y = x + 1, given by the quotient.

Q: How can I check my division result is correct?

Multiply the divisor (x^2-1) by the quotient (x+1) and add the remainder (2x+3); the result should equal the original numerator x^3+x^2+x+2.

Q: How would you integrate this rational function?

Integrate x+1 to get x^2/2 + x. For remainder use partial fractions: (5/2)ln|x-1| − (1/2)ln|x+1|. Sum: x^2/2 + x + (5/2)ln|x-1| − (1/2)ln|x+1| + C.

Q: What if the numerator degree were lower than the denominator?

Then the fraction is proper and no division is needed. You would directly do partial fraction decomposition (if factorable) or other rational-integration/simplification methods.

Answer

Divide leading terms.
x^3 divided by x^2 is x. Multiply (x^2 – 1) by x to get x^3 – x.
Subtract from the dividend.
\[ (x^3 + x^2 + x + 2) – (x^3 – x) = x^2 + 2x + 2 \]
Divide the next term.
x^2 divided by x^2 is 1. Multiply (x^2 – 1) by 1 to get x^2 – 1.
Subtract again.
\[ (x^2 + 2x + 2) – (x^2 – 1) = 2x + 3 \]
State the final result.
The quotient is x + 1 and the remainder is 2x + 3.

\[ \frac{x^3+x^2+x+2}{x^2-1} = x+1 + \frac{2x+3}{x^2-1} \]

Detailed Explanation

We perform the polynomial long division of the dividend \(x^{3}+x^{2}+x+2\) by the divisor \(x^{2}-1\). Steps are given separately and explained in detail.

Set up the division. We divide \(x^{3}+x^{2}+x+2\) by \(x^{2}-1\).
Step 1 — divide the leading terms. Divide the leading term of the current dividend, \(x^{3}\), by the leading term of the divisor, \(x^{2}\).

The quotient term is

\(x^{3}\div x^{2}=x\).

Multiply the divisor by this quotient term:

\(x\cdot(x^{2}-1)=x^{3}-x\).

Subtract this product from the original dividend (subtract term-by-term):

\((x^{3}+x^{2}+x+2)-(x^{3}-x)\).

Compute the subtraction:

\(x^{3}+x^{2}+x+2 – x^{3} + x = x^{2}+2x+2\).

After this subtraction the new working dividend (the remainder so far) is \(x^{2}+2x+2\).
Step 2 — divide the leading terms again. Divide the leading term of the new dividend, \(x^{2}\), by the leading term of the divisor, \(x^{2}\).

The next quotient term is

\(x^{2}\div x^{2}=1\).

Multiply the divisor by this quotient term:

\(1\cdot(x^{2}-1)=x^{2}-1\).

Subtract this product from the current dividend:

\((x^{2}+2x+2)-(x^{2}-1)\).

Compute the subtraction:

\(x^{2}+2x+2 – x^{2} +1 = 2x+3\).

The new remainder is \(2x+3\).
Stop condition. The remainder \(2x+3\) has degree 1, which is less than the degree of the divisor \(x^{2}-1\) (degree 2). Therefore we stop the division.
Collect the result. The quotient is \(x+1\) and the remainder is \(2x+3\). Thus

\[\frac{x^{3}+x^{2}+x+2}{x^{2}-1}=x+1+\frac{2x+3}{x^{2}-1}.\]

Long-division layout (aligned):

       x + 1
   -----------------
x^2 - 1 ) x^3 + x^2 + x + 2
          x^3      - x
         ----------------
               x^2 + 2x + 2
               x^2      - 1
             ----------------
                     2x + 3

See full solution

Try our AI homework help tools

AI for homework

FAQ

What is the quotient and remainder of (x^3 + x^2 + x + 2) divided by (x^2 - 1)?

Quotient = x + 1, remainder = 2x + 3. So (x^3+x^2+x+2)/(x^2-1) = x+1 + (2x+3)/(x^2-1).

How do you perform the long division steps?

Divide leading terms (x^3/x^2 = x), multiply divisor by x, subtract, bring down terms, repeat: add 1, multiply, subtract. Stop when remainder degree < divisor degree.

Can I use synthetic division here?

Not the standard synthetic division: it works for linear divisors x - c. For quadratic divisor use polynomial long division or generalized synthetic method.

Can the remainder be simplified using factorization of the denominator?

Yes: x^2-1 = (x-1)(x+1). Decompose (2x+3)/(x^2-1) = A/(x-1) + B/(x+1) with = 5/2, B = −1/2.

What are the vertical and oblique (slant) asymptotes?

Vertical asymptotes at x = 1 and x = −1 (no cancellations). Oblique asymptote is y = x + 1, given by the quotient.

How can I check my division result is correct?

Multiply the divisor (x^2-1) by the quotient (x+1) and add the remainder (2x+3); the result should equal the original numerator x^3+x^2+x+2.

How would you integrate this rational function?

Integrate x+1 to get x^2/2 + x. For remainder use partial fractions: (5/2)ln|x-1| − (1/2)ln|x+1|. Sum: x^2/2 + x + (5/2)ln|x-1| − (1/2)ln|x+1| + C.

What if the numerator degree were lower than the denominator?