Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Find each quotient when \(P(x)\) is divided by the specified binomial. $$P(x)=3 x^{3}-11 x^{2}-20 x+3 ; \quad x-5$$

Step-by-Step Solution

Answer

The quotient is \(3x^2 + 4x\) with a remainder of 3.

Step 1: Identify the Dividend and Divisor

In this exercise, the dividend is the polynomial \( P(x) = 3x^3 - 11x^2 - 20x + 3 \), and the divisor is the binomial \( x - 5 \). We will perform synthetic division since the divisor is in the form \( x - c \).

Step 2: Set Up Synthetic Division

Write down the coefficients of the polynomial \( P(x) \), which are \( 3, -11, -20, \) and \( 3 \). Write the value of \( c = 5 \) from \( x - 5 \) to the left.

Step 3: Perform Synthetic Division

1. Bring down the leading coefficient (3) to the bottom row.2. Multiply \( 5 \) (value of \( c \)) by \( 3 \) (the number just written down), add this to the next coefficient \( -11 \). Write the result \( 4 \) underneath.3. Repeat: Multiply \( 5 \) by \( 4 \), add to \( -20 \). Write \( 0 \) below.4. Repeat: Multiply \( 5 \) by \( 0 \), add to \( 3 \). Write \( 3 \) below.

Step 4: Write the Quotient and Remainder

The numbers at the bottom represent the coefficients of the quotient polynomial, from left to right, and the last number is the remainder. Therefore, the quotient is \( 3x^2 + 4x + 0 \) with a remainder of 3.

Step 5: State the Quotient

Thus, the quotient of \( P(x) \) divided by \( x - 5 \) is \( 3x^2 + 4x \), and the remainder is 3. The answer can be expressed as \( 3x^2 + 4x + \frac{3}{x-5} \).

Key Concepts

Polynomial Division

To understand polynomial division, you can relate it to the division of numbers that you learned in elementary school. Here, instead of dividing numbers, we divide polynomials. The main goal is to determine how many times a divisor, which is another polynomial, can divide into a given dividend polynomial.

This process results in a quotient and sometimes, a remainder. Polynomial division can be done using different methods, such as synthetic division or polynomial long division. Both methods aim to simplify the polynomial into more manageable pieces.

Quotient and Remainder

In any division process, the quotient and remainder are key components. A quotient is the result of the division, indicating how many times the divisor fits into the dividend. The remainder is what's left after the division is complete.

When dealing with polynomial division, these components tell us:

• The quotient is the polynomial we're left with when the divisor has been evenly divided into the dividend.
• The remainder is what's left when the division isn't perfect.

Knowing the quotient and remainder allows us to express the divided polynomial in a new form: \[\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}\]. This can also be written as a fraction where the remainder is over the original divisor.

Binomial Divisor

A binomial is a two-term expression, such as \(x - 5\). When a polynomial is divided by a binomial, methods like synthetic division become highly effective.

This is because synthetic division is specifically designed for divisors in the form \(x - c\). It simplifies the process by using the constant term from the divisor, allowing for a streamlined division process.

In the exercise given, the binomial divisor was \(x - 5\). Here, the \(c\) value is 5, which simplifies calculations and helps easily determine the quotient and remainder.

Polynomial Long Division

Polynomial long division is a step-by-step method similar to long division with numbers. It is vital for dividing polynomials when synthetic division is not applicable.

The process involves:

• Dividing the first term of the dividend by the first term of the divisor.
• Multiplying the entire divisor by the resulting quotient term and subtracting it from the dividend.
• Repeating these steps with the new polynomial formed, until the remainder is of a lower degree than the divisor.

Polynomial long division provides a comprehensive approach when dividing complex polynomials or when divisors are not in the form needed for synthetic division.