- Rewrite the dividend as \(P(x) = 5x^4 + 0x^3 - 3x^2 + 0x + 4\).
- Apply synthetic division with the root \(x = -1\).
- Identify the quotient \(Q(x) = 5x^3 - 5x^2 + 2x - 2\) and the remainder \(R = 6\).
- The final expression is \(5 x^3-5 x^2+2 x-2+\frac{6}{x+1}\), which matches option b.
- \(\boxed{b}\)
Explanation
- Analysis of the Polynomial Division Problem
The problem asks us to perform the polynomial division of the dividend \(P(x) = 5x^4 - 3x^2 + 4\) by the divisor \(D(x) = x + 1\). We can represent the dividend in descending powers of \(x\) by including missing terms as \(P(x) = 5x^4 + 0x^3 - 3x^2 + 0x + 4\). Our goal is to express this division as \(Q(x) + \frac{R}{D(x)}\), where \(Q(x)\) is the quotient and \(R\) is the constant remainder.
- Performing Synthetic Division
We can use synthetic division to divide the polynomial by the linear factor \((x + 1)\). The root of the divisor \(x + 1 = 0\) is \(x = -1\). Setting up the synthetic division with the coefficients of \(P(x)\) (5, 0, -3, 0, 4), we proceed as follows:
- Bring down the leading coefficient 5.
- Multiply 5 by -1 to get -5, add to 0 to get -5.
- Multiply -5 by -1 to get 5, add to -3 to get 2.
- Multiply 2 by -1 to get -2, add to 0 to get -2.
- Multiply -2 by -1 to get 2, add to 4 to get 6.
The resulting coefficients for the quotient are 5, -5, 2, and -2, and the remainder is 6.
- Comparing with Given Options
From the synthetic division, the quotient polynomial \(Q(x)\) corresponds to the coefficients 5, -5, 2, and -2, which gives \(Q(x) = 5x^3 - 5x^2 + 2x - 2\). The remainder is 6. Therefore, the division \( (5x^4 - 3x^2 + 4) \div (x + 1) \) is equivalent to:
\[ 5x^3 - 5x^2 + 2x - 2 + \frac{6}{x + 1} \]
Comparing this result with the given choices:
A\(5 x^3-8 x^2+8-\frac{4}{x+1}\)
B\(5 x^3-5 x^2+2 x-2+\frac{6}{x+1}\)
C\(5 x^2-5 x+2+\frac{2}{x+1}\)
D\(5 x^2+2 x+\frac{2}{x+1}\)
The calculated result matches option b.
Examples
Polynomial division is a fundamental tool in algebra used for factoring expressions, finding roots, and analyzing functions. For instance, in engineering, when designing systems like filters or control circuits, engineers often need to simplify transfer functions represented by complex ratios of polynomials. By performing polynomial division, they can break down these systems into a more manageable polynomial component and a simpler fractional part, helping to identify key characteristics like stability or resonance.