Problem 21 - A Graphical Approach to Precalculus with Limits

Question

Find each quotient when \(P(x)\) is divided by the specified binomial. $$P(x)=x^{3}+2 x^{2}-17 x-10 ; \quad x+5$$

Step-by-Step Solution

Answer

Quotient is \(x^2 - 3x - 2\).

Step 1: Set up Synthetic Division

First, set up a synthetic division to divide the polynomial \(P(x) = x^3 + 2x^2 - 17x - 10\) by the binomial \(x + 5\). Write the coefficients \([1, 2, -17, -10]\) in a row. The divisor is \(x+5\), so we use the root \(-5\) for synthetic division.

Step 2: Bring Down the First Coefficient

Simply bring down the first coefficient \(1\) (the coefficient of \(x^3\)) to the bottom row.

Step 3: Multiply and Add

Multiply the root \(-5\) by the number just written to the bottom row \(1\), which gives \(-5\). Then, add this result to the next coefficient \(2\) to get \(-3\). Write \(-3\) below the line.

Step 4: Repeat Multiply and Add

Repeat the multiply and add process. Multiply \(-5\) by \(-3\) to get \(15\). Add \(15\) to \(-17\) to get \(-2\). Write \(-2\) below the line.

Step 5: Final Multiply and Add

Again, multiply \(-5\) by \(-2\) to get \(10\). Add \(10\) to \(-10\) to get \(0\). Write \(0\) below the line. A remainder of \(0\) means the division is exact.

Step 6: Write the Quotient Polynomial

The numbers on the bottom row, except for the remainder, now represent the coefficients of the quotient polynomial. The division gives the quotient polynomial: \(x^2 - 3x - 2\).

Key Concepts

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a binomial of the form \(x - c\), where \(c\) is a constant. This technique is beneficial because it reduces the complexity associated with polynomial long division, making calculations quicker and easier. The process involves

• Using the coefficients of the polynomial you want to divide.
• Applying a root of the divisor to perform straightforward multiplication and addition operations.

First, list out the coefficients of your polynomial \(P(x) = x^3 + 2x^2 - 17x - 10\), giving us \([1, 2, -17, -10]\). Since the divisor is \(x + 5\), we use \(-5\) as the root. Set up a synthetic division table by writing these coefficients in a row. Through a sequence of multiplication and addition using the root \(-5\), you arrive at new values that help determine the quotient and any remainder.

Quotient Polynomial

The quotient polynomial is the resulting polynomial when conducting a division between two polynomials. It's the answer obtained from the division process, lacking any remainder part. In our example, after performing synthetic division, we obtained the coefficients \([1, -3, -2]\) at the bottom row.

• The quotient polynomial uses these coefficients to define its terms.
• For our exercise, these translate to \(x^2 - 3x - 2\).

This quotient polynomial represents the reduced degree polynomial obtained after "dividing out" the given binomial \(x + 5\). It's lower in degree by one, compared to the original polynomial \(P(x)\).

Remainder

In polynomial division, the remainder is what is left over after dividing the polynomial completely by the binomial. Ideally, when a polynomial divides evenly, the remainder is zero. Performing synthetic division here, after multiply-and-add steps:

• We multiply \(-5\) by the completed coefficient to obtain new values, ending with a final value in the bottom row.
• Our final outcome was a remainder of \(0\).

A remainder of zero indicates that \(P(x)\) is exactly divisible by \(x+5\), confirming our calculations. In cases where there is a non-zero remainder, it would be expressed as a fractional part of the division answer or added to the final result as \(\text{remainder}/\text{divisor}\).

Polynomial Coefficients

Polynomial coefficients are the numerical factors of a polynomial's terms. They tell you how much of each power of \(x\) is present in the polynomial. For example, in the polynomial \(x^3 + 2x^2 - 17x - 10\), the coefficients are \([1, 2, -17, -10]\). Here is a quick breakdown:

• The coefficient for \(x^3\) (highest degree term) is \(1\).
• For \(x^2\), the coefficient is \(2\).
• For \(x\), it is \(-17\).
• The constant term is \(-10\).

These coefficients are crucial when setting up synthetic division, as they will be used to determine the resulting quotient and remainder. By applying synthetic division iteratively, they are transformed into coefficients of the quotient polynomial.