Answered

Find the quotient.

$\left(5 x^4-3 x^2+4\right) \div(x+1)$


A$5 x^3-5 x^2+2 x-2+\frac{6}{x+1}$
B$5 x^3-8 x^2+8-\frac{4}{x+1}$
C$5 x^2-5 x+2+\frac{2}{x+1}$
D$5 x^2+2 x+\frac{2}{x+1}$

Verified Answer :

  • Represent the dividend as \(5x^4 + 0x^3 - 3x^2 + 0x + 4\).
  • Perform long division by the divisor \(x + 1\).
  • Obtain the quotient \(5x^3 - 5x^2 + 2x - 2\) and the remainder \(6\).
  • State the final expression as \(\boxed{5x^3 - 5x^2 + 2x - 2 + \frac{6}{x + 1}}\).

Explanation

  1. Setting Up the Division

To divide the polynomial \(P(x) = 5x^4 - 3x^2 + 4\) by the divisor \(D(x) = x + 1\), we can express the polynomial in its complete form, including the terms with zero coefficients:
\[P(x) = 5x^4 + 0x^3 - 3x^2 + 0x + 4.\]
The division algorithm states that we can write the dividend as \(P(x) = D(x)Q(x) + R\), where \(Q(x)\) is the quotient and \(R\) is the constant remainder. We will use polynomial long division to find these.

  1. Executing the Division

Performing the long division:

  1. Divide the first term of the dividend \(5x^4\) by the first term of the divisor \(x\), which gives \(5x^3\).
  2. Multiply \(5x^3\) by \((x+1)\) to get \(5x^4 + 5x^3\).
  3. Subtract this from the original polynomial: \((5x^4 + 0x^3 - 3x^2 + 0x + 4) - (5x^4 + 5x^3) = -5x^3 - 3x^2 + 0x + 4\).
  4. Divide \(-5x^3\) by \(x\), which gives \(-5x^2\).
  5. Multiply \(-5x^2\) by \((x+1)\) to get \(-5x^3 - 5x^2\).
  6. Subtract this: \((-5x^3 - 3x^2 + 0x + 4) - (-5x^3 - 5x^2) = 2x^2 + 0x + 4\).
  7. Divide \(2x^2\) by \(x\), which gives \(2x\).
  8. Multiply \(2x\) by \((x+1)\) to get \(2x^2 + 2x\).
  9. Subtract this: \((2x^2 + 0x + 4) - (2x^2 + 2x) = -2x + 4\).
  10. Finally, divide \(-2x\) by \(x\), which gives \(-2\).
  11. Multiply \(-2\) by \((x+1)\) to get \(-2x - 2\).
  12. Subtract this: \((-2x + 4) - (-2x - 2) = 6\).

The resulting quotient is \(Q(x) = 5x^3 - 5x^2 + 2x - 2\) and the remainder is \(R = 6\).

  1. Conclusion

Combining the quotient and the remainder, the result of the division is:
\[\frac{5x^4 - 3x^2 + 4}{x+1} = 5x^3 - 5x^2 + 2x - 2 + \frac{6}{x + 1}.\]
Comparing this result with the given options, we see that it matches the first option provided.

Examples

Polynomial division is a fundamental tool in algebra used for simplifying complex expressions and solving equations. In physics, for instance, when analyzing the motion of an object under a variable force, you might encounter equations where you need to divide a higher-degree polynomial by a lower-degree one to find an simpler expression for velocity or position. Mastering this skill allows for more efficient modeling and problem-solving in various scientific and engineering fields.

Related Questions