Verified Answer :
- Prepare the polynomial \(x^3 + 0x^2 - 20x + 15\) with coefficients \(1, 0, -20, 15\) and divisor \(x-4\) with \(c=4\).
- Apply synthetic division: bring down 1, multiply by 4 to get 4, add to 0 to get 4, multiply by 4 to get 16, add to -20 to get -4, multiply by 4 to get -16, add to 15 to get -1.
- Identify the quotient \(q(x) = x^2 + 4x - 4\) and remainder \(r = -1\).
- Write the final result as \(\boxed{x^2 + 4x - 4 - \frac{1}{x-4}}\).
Explanation
- Problem Analysis and Preparation
We need to perform the division of the polynomial \(P(x) = x^3 - 20x + 15\) by the linear binomial \(b(x) = x - 4\). Synthetic division is an efficient method for dividing a polynomial by a linear term of the form \(x - c\). In our case, the dividend is \(x^3 + 0x^2 - 20x + 15\), where we include a \(0x^2\) term to account for the missing squared term. The divisor is \(x - 4\), so \(c = 4\). The coefficients of our dividend are \(1\), \(0\), \(-20\), and \(15\).
- Performing Synthetic Division
We set up the synthetic division using the value \(c = 4\) and the coefficients of the polynomial.
- Bring down the first coefficient, which is \(1\).
- Multiply this by \(c=4\) to get \(1 \times 4 = 4\). Add this to the next coefficient (0): \(0 + 4 = 4\).
- Multiply the new result by \(c=4\) to get \(4 \times 4 = 16\). Add this to the next coefficient (-20): \(-20 + 16 = -4\).
- Multiply this result by \(c=4\) to get \(-4 \times 4 = -16\). Add this to the last coefficient (15): \(15 - 16 = -1\).
The resulting numbers at the bottom are \(1, 4, -4\) for the quotient and \(-1\) for the remainder.
- Forming the Result
The values we calculated, \(1, 4, -4\), represent the coefficients of the quotient polynomial \(q(x)\) of degree \(n-1\), where \(n\) is the degree of the original polynomial. Since the original polynomial was degree 3, the quotient is degree 2:
\(q(x) = 1x^2 + 4x - 4\).
The final value obtained, \(-1\), is the remainder \(r\).
Expressing the result as \(q(x) + \frac{r}{b(x)}\), we get:
\(x^2 + 4x - 4 + \frac{-1}{x - 4}\).
Examples
Synthetic division is a shorthand method for polynomial division that is frequently used in calculus and algebra when solving for the roots of polynomials. For instance, when engineers analyze the stability of a control system, they often need to divide a characteristic polynomial by a known root to simplify the equation. This helps them determine the remaining poles of the system efficiently, just as we simplified the division of \(x^3 - 20x + 15\) by \(x - 4\) to find the quotient \(x^2 + 4x - 4\) and remainder \(-1\).