Problem 23 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Divisor and Dividend

The problem provides us with a polynomial \(P(x) = 3x^3 - 11x^2 - 20x + 3\) and the divisor \(x - 5\). Our task is to divide \(P(x)\) by \(x - 5\).

Step 2: Set Up Synthetic Division

We use synthetic division to divide a polynomial by a binomial of the form \(x - c\) where \(c = 5\). Write the coefficients of \(P(x)\): 3, -11, -20, and 3. To the left, write 5 (from \(x - 5\)).

Step 3: Perform Synthetic Division

1. Bring down the first coefficient (3) as is. 2. Multiply it by 5 (the divisor) and write the result under the next coefficient: - 5 * 3 = 15. 3. Add \(-11 + 15 = 4\). 4. Repeat this: multiply by 5, write the result under the next coefficient, and add: - 5 * 4 = 20; then \(-20 + 20 = 0\). - 5 * 0 = 0; then \(3 + 0 = 3\).

Step 4: Interpret the Results

The result of the synthetic division gives us the coefficients of the quotient. The quotient is \(3x^2 + 4x + 0\) or \(3x^2 + 4x\). The remainder is 3.

Step 5: Write the Final Expression

The division of \(P(x)\) by \(x-5\) is expressed as \(P(x) = (x-5)(3x^2 + 4x) + 3\). Therefore, the quotient is \(3x^2 + 4x\) and the remainder is 3.

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another. Just like how we divide numbers, this process involves breaking down polynomials into simpler components.
In essence, polynomial division answers the question: "How many times does a given polynomial fit into another?" There are two primary types of polynomial division:

• Long Division: Similar to numerical long division, used for dividing any polynomial by another.
• Synthetic Division: A shortcut method applicable when dividing a polynomial by a binomial of the form \(x - c\).

For our exercise, we utilize synthetic division as it is faster and less complex, especially when the divisor is a simple linear polynomial. It helps us easily find the quotient and remainder when dividing polynomials.

Remainder Theorem

The Remainder Theorem provides a quick way to evaluate the remainder of a polynomial division without performing full division.
It states that when dividing a polynomial \(P(x)\) by a binomial of the form \(x-c\), the remainder is \(P(c)\). This means you substitute \(c\) into the polynomial to get the remainder.
This theorem becomes very handy when you only need the remainder and not the entire quotient. However, in our exercise, synthetic division efficiently gives us both the quotient and the remainder simultaneously.
For example, dividing \(P(x) = 3x^3 - 11x^2 - 20x + 3\) by \(x-5\), we could substitute \(x = 5\) into \(P(x)\) to check the remainder, which would also be 3, confirming the synthetic division result.

Binomial Divisor

A binomial divisor is a polynomial with two terms. In the context of polynomial division, a binomial divisor usually takes the form \(x-c\).
In our exercise, the binomial divisor is \(x-5\). This is simple enough to allow synthetic division, making it a popular choice for simplifying polynomial division problems.
Using synthetic division with a binomial divisor is streamlined and involves:

• Extracting the constant \(c\) from the binomial \(x - c\).
• Applying synthetic division rules, which are faster given the format.

Binomial divisors let students swiftly determine both the quotient and remainder, reinforcing understanding of division even in algebraic terms.

Quotient and Remainder

In polynomial division, the quotient is what you get when the dividend is divided by the divisor, similarly to regular numerical division.
The remainder is what's left over after the division is complete. Just like dividing numbers, a polynomial division yields a result consisting of fully fitting parts (the quotient) and leftover parts (the remainder).
For example, in our exercise:

• The polynomial \(P(x)\) divided by \(x-5\) results in the quotient \(3x^2 + 4x\).
• The remainder of this division is simply 3.

This final expression, \(P(x) = (x-5)(3x^2 + 4x) + 3\), illustrates how the original polynomial can be rewritten using its divisor, quotient, and remainder. This practice solidifies an understanding of how pieces fit into an algebraic puzzle, much like piecing together a mathematical jigsaw.