Problem 47 - A Graphical Approach to College Algebra

Question

Use synthetic division to determine whether the given number is a zero of the polynomial. $$\text { 4; } P(x)=2 x^{3}-6 x^{2}-9 x+6$$

Step-by-Step Solution

Answer

4 is not a zero of the polynomial.

Step 1: Set up Synthetic Division

First, set up the synthetic division. Write down the coefficients of the polynomial \(P(x) = 2x^3 - 6x^2 - 9x + 6\). These coefficients are 2, -6, -9, and 6. Place the number 4, which is to be tested as a zero, outside the synthetic division bracket.

Step 2: Initial Division Process

Bring down the first coefficient, which is 2, as it is. This is the beginning of our work column for synthetic division.

Step 3: Multiply and Add

Multiply the 4 (the potential zero) by 2 (the number you just brought down). Write the result, 8, underneath the next coefficient, which is -6. Add -6 and 8 together to get 2. Write this 2 in the result line.

Step 4: Continue the Division

Repeat the multiplication: Multiply 4 by the new result, 2. Write the result, 8, under the next coefficient, -9. Add -9 and 8 to get -1. Write -1 in the result line.

Step 5: Final Calculation

Finally, multiply 4 by the result of -1. The result is -4. Write this value under the last coefficient, 6. Add 6 and -4 to get 2, which will be the remainder.

Step 6: Analyze Result

Since the remainder is 2 and not 0, 4 is not a zero of the polynomial \(P(x) = 2x^3 - 6x^2 - 9x + 6\).

Key Concepts

Polynomial Division

Polynomial division is a technique that is used to divide one polynomial by another, similar to how you might perform long division with numbers. In this process, a polynomial (the dividend) is divided by another polynomial (the divisor) to get a quotient and sometimes a remainder.
There are different methods for performing polynomial division:

• Long Division: This is similar to regular arithmetic long division, but it can be complex and time-consuming.
• Synthetic Division: This is a shortcut method used specifically when the divisor is of the form \(x - c\). It's much quicker than long division.

The main advantage of synthetic division is its simplicity and speed, especially when dealing with polynomials in a particular form. It organizes the division process in a way that is easier to manage by focusing only on the coefficients rather than the entire expressions. This allows for quicker calculations and is especially useful when checking if a given number is a root of a polynomial.

Zero of a Polynomial

A zero of a polynomial is a value of \(x\) that, when substituted into the polynomial, results in a value of zero. In simpler terms, it's the point where the polynomial crosses the x-axis on a graph.
Finding zeros is crucial because:

• They help in understanding the behavior of the polynomial.
• Zeros can determine the shape and direction of the graph of the polynomial.
• They are solutions to the equation \(P(x) = 0\).

When using synthetic division, a number is a zero of the polynomial if the remainder of the division is zero. In other words, performing synthetic division with a potential zero helps confirm if that value is indeed a zero.

Remainder Theorem

The Remainder Theorem is a principle that connects the result of a polynomial division with its evaluation at a specific point. Specifically, when a polynomial \(P(x)\) is divided by \(x - c\), the remainder of this division is \(P(c)\). This means that you can find the remainder simply by plugging the value of \(c\) into the polynomial.
Here's why this theorem is useful:

• Quickly check if a number is a zero of a polynomial: If \(P(c) = 0\), then \(c\) is a zero.
• Aids in factoring polynomials: Knowing the zeros helps in breaking down polynomials into their factors.

In practical terms, as demonstrated in synthetic division from the original problem, since the remainder was 2 (not zero), it confirmed that 4 is not a zero of the polynomial. The Remainder Theorem gives a straightforward way to understand and verify this outcome.