Question

Use synthetic division to determine whether \(x-\) c is a factor of the given polynomial. \(3 x^{4}-x^{3}+6 x-2 ; \quad x-\frac{1}{3}\)

Short answer

\(x - \frac{1}{3}\) is not a factor.

Step-by-step solution

Set up the synthetic division

First, write down the coefficients of the polynomial in descending order of powers: \[3, -1, 0, 6, -2\]Since \( x - c \) is \( x - \frac{1}{3} \), the divisor to use in the synthetic division is \( \frac{1}{3} \).

Perform synthetic division

Write the divisor on the left and the coefficients in a row:\[ \begin{array}{r|rrrrr} \frac{1}{3} & 3 & -1 & 0 & 6 & -2 \ \ \end{array} \]Bring down the first coefficient (3) directly below the line. Next, multiply \(3\) by \( \frac{1}{3} \). Write the result (1) below the second coefficient and add it to the second coefficient (-1), giving 0.

Continue the process

Continue with the result (0) from the previous step. Multiply \(0\) by \( \frac{1}{3} \). Write the result (0) below the third coefficient (0) and add it, giving 0.Repeat these steps for the remaining coefficients:\(6 \times \frac{1}{3} = 2\)Add 2 to the fourth coefficient (6), resulting in 8.\(-2 \times \frac{1}{3} = -\frac{2}{3} \)Add this to the last coefficient (-2), getting \(-2 - \frac{2}{3} = -\frac{8}{3}\).

Analyze the remainder

The final row of numbers after performing synthetic division gives:\[3, 0, 0, 8, -\frac{8}{3}\]The last number \( -\frac{8}{3} \) is the remainder. Since the remainder is not zero, \( x - \frac{1}{3} \) is not a factor of the polynomial.

Key concepts

Polynomial Factorization

When working with polynomials, one common task is to break them down into simpler components, or factors. This process is called polynomial factorization. It helps in solving polynomial equations and simplifying expressions.

To factorize a polynomial, you're looking for polynomials of lower degrees that multiply together to give the original polynomial. For instance, if we have a polynomial like \(3x^4 - x^3 + 6x - 2\), we're trying to find polynomials of smaller degrees such that their product equals \(3x^4 - x^3 + 6x - 2\).

Factoring polynomials can be challenging. It's easier for simpler polynomials but becomes complex with higher degrees and more terms. Techniques like synthetic division, long division, and factoring by grouping are often used. Synthetic division is particularly useful when dealing with linear divisors, like \(x - c\).

Remainder Theorem

The Remainder Theorem is a handy tool in polynomial algebra. It states that if you divide a polynomial \(f(x)\) by a linear divisor \(x - c\), the remainder is simply \(f(c)\).

For example, if you have the polynomial \(3x^4 - x^3 + 6x - 2\) and you want to check if \(x - \frac{1}{3}\) is a factor, you can use the Remainder Theorem. Plug \(c = \frac{1}{3}\) into the polynomial and evaluate it. If \(f(\frac{1}{3}) = 0\), then \(x - \frac{1}{3}\) is a factor.

Using synthetic division as outlined in the solution:

• Set up coefficients and divisor (\(\frac{1}{3}\)).
• Perform division steps.
• The remainder we found was \(-\frac{8}{3}\).

Because the remainder is not zero, \(x - \frac{1}{3}\) is not a factor of the polynomial.

Divisor in Polynomial Division

In polynomial division, the divisor is the polynomial by which you are dividing. If you're using synthetic division, the divisor will be in the form of \(x - c\).

In our example: \(3x^4 - x^3 + 6x - 2\) divided by \(x - \frac{1}{3}\). The divisor here is \(x - \frac{1}{3}\). You use this value to set up your synthetic division: \(\frac{1}{3}\).

Here's the process in steps:

• Write down the divisor \(\frac{1}{3}\).
• List coefficients of the polynomial.
• Bring down the first coefficient.
• Multiply the divisor by the last number in the row and add it to the next coefficient.
• Repeat until you've worked through all coefficients.
• The final number is your remainder.

By following these steps, you can determine if a given \(x - c\) is a factor of the polynomial. If the remainder is zero, \(x - c\) is a factor. If not, it isn't.