Problem 52 - A Graphical Approach to College Algebra

Question

Use synthetic division to determine whether the given number is a zero of the polynomial. $$-4 ; \quad P(x)=6 x^{3}+25 x^{2}+3 x-3$$

Step-by-Step Solution

Answer

-4 is not a zero of the polynomial.

Step 1: Set Up the Synthetic Division

To set up synthetic division, write the coefficients of the polynomial \( P(x) = 6x^3 + 25x^2 + 3x - 3 \). The coefficients are 6, 25, 3, and -3. We will divide by \( x + 4 \), so use -4 for the synthetic division.

Step 2: Bring Down the Leading Coefficient

Write down the leading coefficient (6) below the division line to start the computation. This is the first coefficient of the synthetic division process.

Step 3: Synthetic Division Process

Multiply the last result by the divisor (-4), write the result under the next coefficient, then add. Repeat this process for all coefficients. Start by multiplying 6 (from under the line) by -4 to get -24, which goes under 25. Add 25 and -24 to get 1.

Step 4: Continue with Synthetic Steps

Multiply 1 (result from previous step) by -4, resulting in -4. Add this to 3 (the next coefficient) to get -1. Continue by multiplying -1 by -4 to get 4 and add this to -3 to get 1.

Step 5: Identify the Remainder

In synthetic division, the last number is the remainder. Since the final number is 1, there is a non-zero remainder, indicating -4 is not a zero of the polynomial.

Key Concepts

Polynomial Division

Polynomial division is similar to long division used with numbers, except that it divides polynomials instead of numbers. It's essential for simplifying polynomial expressions and finding solutions. There are two main types of polynomial division:

• Long division method: Works very much like the numerical long division but can be cumbersome with higher-degree polynomials.
• Synthetic division: A simplified method that is faster and more straightforward, but only works when dividing by linear divisors, typically of the form \(x - c\).

When using synthetic division, you only need the coefficients of the polynomial, greatly simplifying calculations. It's a handy tool to determine roots or potential zeros of polynomials. Synthetic division is particularly efficient for checking polynomial roots because it sidesteps much of the complexity of writing out all the polynomial terms.

Zeros of Polynomial

The zeros of a polynomial, also known as roots or solutions, are the values of \(x\) that make the polynomial equal to zero. Determining the zeros is a key aspect of solving polynomial equations.

• If \((x - c)\) is a factor of the polynomial, then \(c\) is a root or zero of the polynomial.
• To test if a number is zero of the polynomial, substitute it into the polynomial equation. If it results in zero, then the number is a valid zero.

Synthetic division can efficiently verify if a particular number is a zero of the polynomial with minimal calculations. If, after synthetic division, there is a remainder of zero, then the tested number is a zero of the polynomial. Hence, in our exercise, after dividing by \(-4\), a remainder of 1 showed that \(-4\) is not a zero for the given polynomial.

Remainder Theorem

The Remainder Theorem is a fundamental principle that links polynomial division to polynomial evaluation. It states that if a polynomial \(P(x)\) is divided by a linear factor \((x - c)\), the remainder of this division is \(P(c)\).

• In simple terms, the remainder of the division tells us what the value of the polynomial is when \(x\) is \(c\).
• If the remainder is zero, \(c\) is a zero or root of the polynomial.

In the context of synthetic division, the last number we get is the remainder. In our exercise, when using \(-4\) as the test zero, the remainder was 1, which showed that \(P(-4) = 1\). Therefore \(-4\) is not a zero, because a zero remainder would mean that \(-4\) perfectly divides the polynomial.