Answered

Use synthetic division to find the result when $3 x^4+15 x^3+3 x^2+16 x+12$ is divided by $x+5$. If there is a remainder, express the result in the form $q(x)+\frac{r(x)}{b(x)}$.

Verified Answer :

  • We use the synthetic division method with the divisor constant \(c = -5\) and the dividend coefficients \(3, 15, 3, 16, 12\).
  • Through the step-by-step multiplication and addition process, we identify the quotient coefficients \(3, 0, 3, 1\) and the remainder \(7\).
  • The resulting quotient is the polynomial \(3x^3 + 3x + 1\).
  • The final expression is \(\boxed{3x^3 + 3x + 1 + \frac{7}{x+5}}\).

Explanation

  1. Analyzing the Problem and Setting Up Synthetic Division

We are tasked with dividing the polynomial \(P(x) = 3x^4 + 15x^3 + 3x^2 + 16x + 12\) by the linear divisor \(D(x) = x + 5\) using the method of synthetic division. Synthetic division is a shorthand method for performing polynomial division when the divisor is a linear binomial of the form \(x - c\). In this case, \(x + 5 = x - (-5)\), so our value for \(c\) is \(-5\). The coefficients of our dividend polynomial \(P(x)\) are \(3, 15, 3, 16, 12\).

  1. Performing the Synthetic Division

To perform synthetic division, we arrange the coefficients of the dividend in a row and use the value \(c = -5\) on the left side:

\[ \begin{array}{c|rrrrr} -5 & 3 & 15 & 3 & 16 & 12 \ & & -15 & 0 & -15 & -5 \\ & 3 & 0 & 3 & 1 & 7 \end{array} \]

Here is the step-by-step breakdown of the calculation:

  1. Bring down the first coefficient: \(3\).
  2. Multiply the value \(-5\) by the current bottom value \(3\) to get \(-15\). Add this to the next coefficient \(15\): \(15 + (-15) = 0\).
  3. Multiply \(-5\) by \(0\) to get \(0\). Add this to the next coefficient \(3\): \(3 + 0 = 3\).
  4. Multiply \(-5\) by \(3\) to get \(-15\). Add this to the next coefficient \(16\): \(16 + (-15) = 1\).
  5. Multiply \(-5\) by \(1\) to get \(-5\). Add this to the final coefficient \(12\): \(12 + (-5) = 7\).

The numbers in the last row are the coefficients of the quotient polynomial \(Q(x)\) and the remainder \(r\). Since the original dividend was a degree-4 polynomial, the quotient will be a degree-3 polynomial. Thus, the coefficients \(3, 0, 3, 1\) correspond to \(3x^3 + 0x^2 + 3x + 1\), and the remainder is \(7\).

  1. Conclusion and Final Form

With the quotient \(Q(x) = 3x^3 + 3x + 1\) and the remainder \(r = 7\), we can write the final result of the division in the requested form:

\[ \frac{3x^4 + 15x^3 + 3x^2 + 16x + 12}{x + 5} = Q(x) + \frac{r}{D(x)} = 3x^3 + 3x + 1 + \frac{7}{x+5}. \]

The final result is \(\boxed{3x^3 + 3x + 1 + \frac{7}{x+5}}\).

Examples

Synthetic division is a highly efficient tool in engineering and computer science, particularly in control systems and digital signal processing. When designing filters or analyzing the stability of a system, engineers often need to divide complex transfer functions represented by polynomials. By using synthetic division, they can quickly simplify these expressions to identify poles, zeros, or the steady-state response of a system without performing long-form polynomial division, which is far more time-consuming and prone to manual calculation errors. It serves as a fundamental building block for algorithmic complexity management in software development as well.

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