Problem 26 - Algebra for College Students

Question

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x-4 \text { a factor of } 2 x^{3}-11 x^{2}+10 x+8 ? $$

Step-by-Step Solution

Answer

Yes, \( x - 4 \) is a factor of the polynomial.

Step 1: Apply the Factor Theorem

The Factor Theorem states that \( x - a \) is a factor of a polynomial \( f(x) \) if and only if \( f(a) = 0 \). We need to check if \( x - 4 \) is a factor by evaluating the polynomial at \( x = 4 \). First, substitute \( x = 4 \) into the polynomial \( f(x) = 2x^3 - 11x^2 + 10x + 8 \).

Step 2: Substitute and Simplify

Substitute \( x = 4 \) into the polynomial: \[ f(4) = 2(4)^3 - 11(4)^2 + 10(4) + 8 \] Calculate each term: \[ 2(4)^3 = 2(64) = 128 \] \[ 11(4)^2 = 11(16) = 176 \] \[ 10(4) = 40 \] Now, add these evaluated terms together: \[ f(4) = 128 - 176 + 40 + 8 \]

Step 3: Calculate the Result

Simplify the expression from the previous step: \[ 128 - 176 = -48 \] \[ -48 + 40 = -8 \] \[ -8 + 8 = 0 \] The result is \( f(4) = 0 \). This means that \( x - 4 \) is indeed a factor of the polynomial because the result is zero as per the Factor Theorem.

Key Concepts

Polynomial Division

Polynomial division is a method used to divide polynomials, much like how you divide numbers. The main operations involved are division and subtraction, repeatedly applied. When you divide one polynomial by another, you may end up with a quotient and sometimes with a remainder. This can be done using long division of polynomials or synthetic division, which is a shortcut method.

• Long Division: Similar to arithmetic division, align the dividend and divisor and see how many times the divisor can "fit into" the larger part of the dividend.
• Synthetic Division: Primarily used when dividing by a binomial of the form \(x-a\), and it is faster than long division for this purpose.

Polynomial division is essential for simplifying expressions and finding polynomial factors, especially when applying the Factor Theorem. Mastering this process can make solving polynomial expressions much easier.

Roots of Polynomials

The roots of a polynomial are the values of \(x\) that make the polynomial equal to zero. This is crucial because it helps identify the factors of the polynomial. If you know the roots, you can easily build an expression for the polynomial, as it will have factors like \((x - r_1), (x - r_2), \ldots\), where \(r_1, r_2, \ldots\) are the roots.

• Finding Roots: One way to find roots is by solving the polynomial equation \(f(x) = 0\).
• Using the Factor Theorem: This theorem helps identify roots by using potential factors, as discussed in the exercise.
• Multiplicity of Roots: Sometimes roots appear multiple times. For example, if a root is repeated three times, it contributes a cube factor \((x - r)^3\) to the polynomial.

Identifying these roots is fundamental in simplifying polynomials and understanding their graph structure, as each root is a point where the polynomial crosses or touches the x-axis.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations ( like addition, subtraction, multiplication, and division). Polynomials are a specific type of algebraic expression involving powers of a variable. Understanding these expressions is key to algebra, as they form the basis for equations and functions.

• Components: Made up of terms, which are divisble into coefficients and variables raised to exponents.
• Simplification: Process where expressions are rewritten in simpler or more practical forms, often using common techniques like factoring and expanding.
• Equations: Algebraic expressions set equal to a value, commonly zero when solving for roots or factors.

Understanding algebraic expressions is necessary for solving polynomial problems, especially when applying the Factor Theorem to identify factors or when simplifying complex expressions for easier manipulation.