Problem 41 - 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. $$ x-2,4 x^{4}-15 x^{2}-4 $$

Step-by-Step Solution

Answer

The remainder is -76, so \(x-2\) is not a factor.

Step 1: Convert Division Problem into Synthetic Form

To perform synthetic division, first ensure the polynomial is in descending order of powers. The divisor is \(x-2\), so we will use 2 (the zero of the divisor) for synthetic division. The polynomial is \(4x^4 + 0x^3 - 15x^2 + 0x - 4\). Note the placeholders for missing degrees.

Step 2: Set Up Synthetic Division Table

Write the zero of the divisor (2) on the left. Write the coefficients of the polynomial: 4, 0, -15, 0, -4, in a row to the right.

Step 3: Perform Synthetic Division

Bring down the first coefficient (4) as is. Multiply 2 by 4 to get 8, and write it under the next coefficient. Add 0 and 8 to get 8. Continue this process: multiply the result by 2, add it to the next coefficient, and write the sum below until completed with all coefficients.

Step 4: Check Remainder for Factorization

The last number in the bottom row is the remainder, which is -76. Since -76 is not 0, \(x-2\) is not a factor of \(4x^4 - 15x^2 - 4\).

Key Concepts

Polynomial Division with Synthetic Division

Polynomial division is a process to divide a polynomial by another polynomial. It is much like long division with numbers and has a method named synthetic division for specific types of polynomials. Synthetic division is an easy and fast way when the divisor is a linear polynomial of the form \(x - c\).

This method simplifies calculations by only using the coefficients of the polynomials. Let's use our example: we have the divisor \(x-2\) and the polynomial \(4x^4 - 15x^2 - 4\).

• First, note that the zero of the divisor \(x-2\) is 2, which we'll use in the synthetic division process.
• The polynomial is made complete by placing zeros where terms are missing, hence \(4x^4 + 0x^3 - 15x^2 + 0x - 4\).

This preparation allows us to set up the synthetic division table. With this, we focus mainly on the operations with coefficients by following a pattern of multiply and add to get the remainder and resulting polynomial.

Factorization with Synthetic Division

Factorization is breaking down a polynomial into simpler "factor" polynomials that can be multiplied together to give the original polynomial. It helps to find zeros of polynomials and solve polynomial equations.

Using synthetic division, factorization is tested by checking whether a divisor leaves a remainder of zero.

• The key here is that if the remainder is zero, then the divisor is a factor of the polynomial.
• For instance, if after performing synthetic division, no remainder is left, it means \(x - c\) is a factor.

For the given polynomial \(4x^4 - 15x^2 - 4\), when using \(x-2\) as a divisor, the remainder turned out to be -76. This indicates \(x-2\) is not a factor of the polynomial, as a zero remainder is a requirement for factorization here.

Understanding the Remainder Theorem

The remainder theorem is a useful concept when dividing polynomials, especially in conjunction with synthetic division.

It states that if a polynomial \(f(x)\) is divided by \(x-c\), the remainder of this division is \(f(c)\).

• In practical terms, it's like plugging \(c\) into the polynomial and seeing what the result is.
• If \(f(c) = 0\), it indicates \(x-c\) is a factor, confirming the factorization.

In our case study, the remainder was -76. Thus, evaluating the polynomial at \(x=2\) confirms this result since it’s the calculated value substituting \(x\) with 2 into \(f(x)\). It clearly reflects that \(x-2\) does not divide the polynomial evenly, aligning with the factorization failure during synthetic division.