Problem 23 - Algebra for College Students

Question

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x+3 \text { a factor of } 6 x^{2}+13 x-14 ? $$

Step-by-Step Solution

Answer

No, \( x+3 \) is not a factor of \( 6x^2 + 13x - 14 \).

Step 1: Understanding the Factor Theorem

The Factor Theorem states that a polynomial \( f(x) \) has a factor \( (x - c) \) if and only if \( f(c) = 0 \). In this problem, we need to verify whether \( x+3 \) is a factor of \( 6x^2 + 13x - 14 \). This means we should check if substituting \( x = -3 \) into the polynomial results in zero.

Step 2: Substitute Value into the Polynomial

Substitute \( x = -3 \) into the polynomial \( f(x) = 6x^2 + 13x - 14 \). Compute \( f(-3) \) as follows:1. Calculate \( 6(-3)^2 = 6 \times 9 = 54 \).2. Calculate \( 13(-3) = -39 \).3. Combine terms to get \( 54 - 39 - 14 \).

Step 3: Simplify the Expression

Now simplify the expression obtained from Step 2:- Begin with \( 54 - 39 = 15 \).- Then compute \( 15 - 14 = 1 \).Since \( f(-3) = 1 \), the polynomial \( 6x^2 + 13x - 14 \) evaluated at \( x = -3 \) does not equal zero.

Step 4: Conclusion

Since \( f(-3) eq 0 \), according to the Factor Theorem, the polynomial \( x+3 \) is not a factor of \( 6x^2 + 13x - 14 \). Therefore, \( x+3 \) does not divide the polynomial evenly.

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, similar to the process of long division with numbers. This technique is essential when working with polynomials to simplify equations or find factors. In polynomial division, the goal is to determine how many times the divisor, a polynomial of lesser or equal degree than the dividend, fits into the dividend without leaving a remainder.

The process involves repeating the following steps:

• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by the result from the first step and subtract it from the dividend.
• Bring down the next term of the dividend if necessary and repeat these steps until a remainder is produced.

Polynomial division helps in simplifying polynomials and finding their factors. When combined with the Factor Theorem, this method allows for checking if a certain term can divide the polynomial without any remainder.

Factors of Polynomials

The factors of a polynomial are expressions that can be multiplied together to obtain the original polynomial. Understanding factors is crucial in algebra, as factoring is often the first step in simplifying expressions or solving polynomial equations.

To identify the factors of a polynomial, one might use:

• The Factor Theorem, which states if a polynomial evaluated at a certain number is zero, then that number is a root, and the polynomial corresponding term is a factor.
• Polynomial division to divide the original polynomial by potential factors to see if they evenly divide it.
• Common roots or known formulas for special cases, like difference of squares or perfect square trinomials.

By finding the factors of polynomials, complex expressions become more manageable, which is particularly valuable in solving equations or sketching graphs.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition or multiplication) that represent specific values. Doing algebraic manipulations requires an understanding of the properties of these expressions and how to perform operations on them effectively.

Key components include:

• Variables: symbols that represent unknown values and can change depending on the context.
• Constants: fixed values that do not change.
• Operations: the mathematical procedures applied to variables and constants, such as addition, subtraction, and division.

Algebraic expressions can be used to solve real-world problems by setting up equations, including polynomial equations. Understanding these expressions is fundamental for solving them or determining the roots and factors, as in the problem where polynomial division and the Factor Theorem are applied to decide if a term is a factor.