Problem 40 - College Algebra

Question

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. $$ x+3,-4 x^{3}+5 x^{2}+8 $$

Step-by-Step Solution

Answer

No, \(x + 3\) is not a factor because the remainder is 161.

Step 1: Set Up Synthetic Division

To use synthetic division for verifying if \(x + 3\) is a factor of \(-4x^3 + 5x^2 + 0x + 8\), rewrite \(x + 3 = 0\) to find the root \(x = -3\). Set up the synthetic division with the coefficients: \(-4, 5, 0, 8\).

Step 2: Perform Synthetic Division

Start the synthetic division by bringing down the leading coefficient \(-4\). Multiply \(-4\) by \(-3\) and add to the next coefficient \(5\). Repeat this process through the polynomial. You will get the results: \(-4\), \(17\), \(-51\), \(161\). The final number (161) is the remainder.

Step 3: Interpret the Result

Since the remainder is not zero, \(x + 3\) is not a factor of \(-4x^3 + 5x^2 + 8\). For \(x + 3\) to be a factor, the remainder must be zero.

Key Concepts

Polynomial Division

Polynomial division is a technique used to divide one polynomial by another. Similar to long division in arithmetic, but specifically designed for polynomials. In arithmetic, you're dividing numbers, while in polynomial division, the focus is on dividing expressions with variables.
Unlike traditional long division, synthetic division is a shortcut method used for dividing a polynomial by a binomial of the form \(x - c\). It simplifies the division process by dealing directly with the coefficients of the polynomials rather than the variables.
This method not only speeds up calculations but also streamlines the process, making it less prone to algebraic errors.

• It's particularly useful in algebra assignments where you have polynomials with large degrees.
• Helps quickly identify whether a particular value is a root of the polynomial.

Factorization

Factorization involves expressing a polynomial as the product of its factors. These factors could include numbers, variables, or smaller polynomials. It's like breaking down a complex expression into simpler, more manageable pieces.

• Through factorization, complex problems become easier to solve or simplify.
• It's commonly taught in algebra to simplify polynomial expressions or solve polynomial equations.

To factor a polynomial, you look for common terms or patterns. Sometimes, using synthetic division can quickly tell us if a binomial is a factor of a polynomial.
For instance, if after using synthetic division, the remainder is zero, then the divisor polynomial is a factor of the given polynomial. However, if there is a remainder, as in our exercise, it indicates that further factorization using the given binomial isn't possible.

Remainder Theorem

The Remainder Theorem is an essential concept that ties directly into the method of synthetic division. It states that if a polynomial \(P(x)\) is divided by a binomial of the form \(x - c\), the remainder of this division is \(P(c)\).
This theorem is a powerful tool because it allows us to quickly determine the value of a polynomial at a particular point without fully factoring it or performing lengthy division.

• If the remainder is zero, then \(x - c\) is a factor of \(P(x)\).
• A non-zero remainder indicates that \(x - c\) is not a factor.

In our case, since dividing \(-4x^3 + 5x^2 + 8\) by \(x + 3\) resulted in a remainder of 161, the theorem confirms that \(x + 3\) is not a factor. This visualization helps directly connect synthetic division results to the theorem, reinforcing the understanding that the remainder reflects whether a simple binomial divides the polynomial evenly.