Verified Answer :

  • Identify the coefficients of the polynomial \(x^3 + 5x^2 + 7x + 8\), which are \(1, 5, 7, 8\), and the divisor's constant \(c = -3\).
  • Bring down the leading coefficient \(1\).
  • Repeatedly multiply the constant \(-3\) by the last calculated value and add it to the next coefficient to obtain the sequence \(1, 2, 1, 5\).
  • Interpret the results: the quotient is \(x^2 + 2x + 1\) and the remainder is \(5\).

Final answer: Quotient: \(\boxed{x^2+2x+1}\), Remainder: \(\boxed{5}\)

Explanation

  1. Problem Analysis and Setup

To perform synthetic division on the polynomial \(P(x) = x^3 + 5x^2 + 7x + 8\) by the binomial \(x + 3\), we first identify the coefficients and the root of the divisor. The divisor \(x + 3\) is in the form \(x - c\), which implies \(c = -3\). The coefficients of the dividend polynomial \(x^3 + 5x^2 + 7x + 8\) are \(1, 5, 7,\) and \(8\). Synthetic division is a simplified way to perform long division of polynomials when dividing by a linear binomial of the form \(x - c\).

  1. Setting up the Synthetic Division Table

We set up the synthetic division table by placing the root \(c = -3\) on the left and the coefficients \(1, 5, 7, 8\) in a row:
\begin{array}{r|rrrr}
-3 & 1 & 5 & 7 & 8 \)
& & & & \)
\
& & & &
\end{array}

  1. Executing Synthetic Division

We perform the division process:

  1. Bring down the first coefficient, \(1\), to the bottom row.
  2. Multiply \(c = -3\) by the value in the bottom row (\(1\)) to get \(-3\). Place this result under the next coefficient (\(5\)). Add the column: \(5 + (-3) = 2\).
  3. Multiply \(c = -3\) by this new value (\(2\)) to get \(-6\). Place this result under the next coefficient (\(7\)). Add the column: \(7 + (-6) = 1\).
  4. Multiply \(c = -3\) by this new value (\(1\)) to get \(-3\). Place this result under the last coefficient (\(8\)). Add the column: \(8 + (-3) = 5\).

The final bottom row values are \(1, 2, 1,\) and \(5\).

  1. Conclusion

The values in the bottom row (excluding the last one) represent the coefficients of the quotient polynomial, which is one degree less than the original polynomial. Since the original polynomial was degree 3, the quotient is degree 2: \(1x^2 + 2x + 1\). The last value, \(5\), is the remainder.

Thus, the quotient is \(x^2 + 2x + 1\) and the remainder is \(5\).
The final result can be expressed as: \(\boxed{x^2+2x+1}\) for the quotient and \(\boxed{5}\) for the remainder.

Examples

Synthetic division is widely used in algebra to quickly find the roots of polynomials, factorize them, or evaluate functions at specific points. For instance, if a company models its profit over time using a polynomial, synthetic division helps analysts quickly check if a specific year results in a profit of zero (the root) or determine the remaining profit structure after accounting for a specific time factor (the linear division). Mastering this technique allows for much faster polynomial manipulation compared to standard long division.

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