Problem 340 - Precalculus

Question

Use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. $$x-2,3 x^{4}-6 x^{3}-5 x+10$$

Step-by-Step Solution

Answer

\(x-2\) is not a factor; remainder is -20.

Step 1: Set Up Synthetic Division

To determine if \(x - 2\) is a factor of \(3x^4 - 6x^3 - 5x + 10\), we'll use synthetic division. Begin by writing the coefficients of the polynomial: \(3, -6, 0, -5, 10\). Notice there is a missing \(x^2\) term, so include a \(0\) for its coefficient. The divisor is \(x - 2\), which means we'll use \(2\) for the synthetic division process.

Step 2: Perform Synthetic Division

Write down the number 2 from \(x-2\) on the left. Bring down the leading coefficient, 3, as is: \[\begin{array}{r|rrrrr}2 & 3 & -6 & 0 & -5 & 10 \ & & 6 & 0 & -10 & -30 \\hline & 3 & 0 & 0 & -15 & -20 \\end{array}\]Multiply 2 by 3, add to the next coefficient (-6), repeat for each pair down the line.

Step 3: Check the Remainder

The result from the synthetic division gives the bottom row of numbers: 3, 0, 0, -15, -20. The last number, -20, is the remainder. Since the remainder is not zero, \(x - 2\) is not a factor of \(3x^4 - 6x^3 - 5x + 10\).

Key Concepts

Polynomial Division

Polynomial division is a method for dividing a polynomial by another polynomial of lesser degree. It's similar to long division of numbers, but with variables and coefficients.
In this method, you start with the highest degree term of the polynomial you are dividing and work your way down. You seek to match the leading term of the divisor (the polynomial you're dividing by) with part of your dividend (the polynomial that's being divided). You repeat the process, subtracting off each calculated term, until you cannot continue.
There are two types of polynomial division: Long Division and Synthetic Division. Long division works with any polynomials, while synthetic division is a simplified version, efficient for dividing by linear polynomials of the form \(x - c\).
Synthetic division is particularly handy because it involves working only with the coefficients, making calculations faster and easier.

Factors of Polynomials

In mathematics, a factor of a polynomial is a polynomial that divides another polynomial without leaving a remainder. When one polynomial is a factor of another, it implies that the division results in another polynomial with remainder zero.
To find if a given polynomial is a factor of another, you perform either long division or synthetic division. If the remainder is zero after division, then the divisor polynomial is indeed a factor.

• Example: If you divide \(x^3 - 6x^2 + 11x - 6\) by \(x - 1\) and the remainder is zero, \(x - 1\) is a factor.
• Factorization involves expressing a polynomial as a product of its factors. For example, \(x^2 - 5x + 6\) can be factored as \((x - 2)(x - 3)\).

Remainder Theorem

The Remainder Theorem is a crucial concept in algebra, stating that the remainder of the division of a polynomial \(f(x)\) by a linear divisor \(x - c\) is equal to \(f(c)\). This theorem is especially useful when using synthetic division.
When you perform synthetic division, and you calculate \(f(c)\) using the coefficients of \(f(x)\), the last number in the bottom row of the synthetic division setup is the remainder.

• If the remainder is zero, then \(x - c\) is a factor of the polynomial.
• If the remainder is not zero, as in our original problem, \(x - c\) is not a factor.

This theorem simplifies the process and helps quickly determine potential factors of polynomials without completing the entire division.