Problem 24 - Precalculus: Functions and Graphs

Question

Use synthetic division to find the quotient and remainder If the first polynomial is divided by the second. $$5 x^{3}-6 x^{2}+15 ; \quad x-4$$

Step-by-Step Solution

Answer

The quotient is \(5x^2 + 14x + 56\) and the remainder is \(239\).

Step 1: Write Down the Divisor and Dividend

Identify and write the dividend as the polynomial to be divided, which is \(5x^3 - 6x^2 + 0x + 15\). Note the placeholder \(0x\) for the missing \(x\) term. The divisor is \(x - 4\).

Step 2: Set up the Synthetic Division

In synthetic division, only the coefficients are used. Write the coefficients of the dividend: \(5, -6, 0, 15\). The divisor \(x - 4\) provides the number \(4\) (change the sign of \(-4\)). Place \(4\) to the left outside of the division symbol.

Step 3: Perform the Synthetic Division

1. Bring down the first coefficient (\(5\)) as it is.2. Multiply \(4\) by \(5\) (get \(20\)) and write this under the next coefficient \(-6\).3. Add \(-6\) and \(20\) to get \(14\).4. Multiply \(4\) by \(14\) (get \(56\)) and write this under the next coefficient \(0\).5. Add \(0\) and \(56\) to get \(56\).6. Multiply \(4\) by \(56\) (get \(224\)) and write this under the next coefficient \(15\).7. Add \(15\) and \(224\) to get \(239\).

Step 4: Interpret the Results

The numbers at the bottom of the division are \(5, 14, 56, 239\). The first three represent the coefficients of the quotient: \(5x^2 + 14x + 56\). The last number \(239\) is the remainder. Therefore, the quotient is \(5x^2 + 14x + 56\) and the remainder is \(239\).

Key Concepts

Polynomial Division

Polynomial division is a critical concept in algebra, especially when dealing with complex polynomials. Unlike long division, synthetic division is a streamlined process used primarily for dividing polynomials by linear expressions of the form \(x - c\). This method focuses only on the coefficients, making it quicker and simpler than traditional division for these specific cases.

There are several advantages to using synthetic division:

• It simplifies the process by avoiding variables until the final step.
• It reduces the potential for errors by focusing on operations with integers or simple numbers, rather than expressions.
• It is especially useful in finding the zeroes or roots of polynomials.

To execute synthetic division, you must first ensure your polynomial is written in its complete form, which includes all degrees of \(x\), even if their coefficients are zero, as seen in the exercise where a placeholder \(0x\) was used to represent the missing term.

Quotient and Remainder

In the context of polynomial division, the quotient and remainder play essential roles in interpreting the results of the division. The quotient is the result of division, a polynomial of one degree less than the original dividend because you are dividing by a linear term \(x - c\).

Here, the given problem divides a cubic polynomial (degree 3) by a linear polynomial (degree 1), resulting in a quotient that is a quadratic polynomial (degree 2). In our example, using synthetic division, the coefficients for the quotient were identified as \(5x^2 + 14x + 56\).

The remainder is the leftover value that cannot be evenly divided by the divisor. For polynomials, this remainder is typically a constant when dividing by a linear factor, which was \(239\) in this case. Understanding the quotient and remainder is vital as they serve as the building block for deeper knowledge in polynomial identities and factorizations.

Coefficients in Polynomials

Coefficients are the numerical part of the terms in a polynomial. In polynomial division, especially synthetic division, managing these coefficients effectively is crucial as they are the primary focus of the operation.

To perform synthetic division:

• Extract the coefficients from each term of the polynomial, including any zero placeholders for missing terms.
• Use addition and multiplication on these coefficients as specified in the synthetic division steps.

The way you manipulate coefficients determines how easily you can achieve the quotient and remainder. In our example, the coefficients of the polynomial \(5x^3 - 6x^2 + 0x + 15\) are \(5, -6, 0,\) and \(15\). After setting up synthetic division with these and the modification provided by the divisor, solving for the new set of coefficients becomes straightforward.

It's essential to understand coefficients because they reflect the changes in the polynomial expressions and are directly connected to the function's behavior. Furthermore, identifying correct coefficients is key to ensuring accuracy when solving algebraic problems through synthetic division.