Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Use the factor theorem to show that \(x-c\) is a factor of \(f(x)\). f(x)=3 x^{3}-x^{2}+5 x-4 ; \quad c=2$

Step-by-Step Solution

Answer

\(x - 2\) is not a factor of \(f(x)\) because \(f(2) \neq 0\).

Step 1: Substitute c into the polynomial

Using the factor theorem, we begin by substituting \( c = 2 \) into the polynomial \( f(x) = 3x^3 - x^2 + 5x - 4 \) to find \( f(2) \). This will help us determine if \( x - 2 \) is a factor of \( f(x) \). Calculate: \[ f(2) = 3(2)^3 - (2)^2 + 5(2) - 4 \] \[ f(2) = 3(8) - 4 + 10 - 4 \] \[ f(2) = 24 - 4 + 10 - 4 \]

Step 2: Simplify the expression

Now, simplify the expression from the previous step to get a single numerical value. \[ f(2) = 24 - 4 + 10 - 4 \] First, calculate \( 24 - 4 = 20 \). Then, \( 20 + 10 = 30 \). Lastly, \( 30 - 4 = 26 \).

Step 3: Analyze the result of f(c)

The value of \( f(2) \) is 26. According to the factor theorem, \( x - c \) is a factor of \( f(x) \) if \( f(c) = 0 \). Since \( f(2) = 26 eq 0 \), \( x - 2 \) is not a factor of \( f(x) \).

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial is \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where \( a_n, a_{n-1}, \ldots, a_0 \) are constants and \( n \) is a non-negative integer which signifies the degree of the polynomial.
Polynomials are very versatile and can represent a wide range of functions. They are not just limited to lines or parabolas but can be crafted to fit curves of all sorts and degrees of complexity.

• The degree of the polynomial indicates the highest power of \( x \) in the expression. A higher degree means the curve can have more bends.
• If the degree is 0, the polynomial is a constant; if it is 1, it is linear; and if it is 2, it is quadratic, and so on.
• In polynomial functions, the coefficients are the numbers in front of the variables (e.g., in \( 3x^3 \), the coefficient is 3).

Polynomials are fundamental in calculus and algebra and are used extensively in both theoretical and applied mathematics.

Factorization

Factorization in mathematics is the process of breaking down an expression into multiple "factors" that, when multiplied together, yield the original expression. In the context of polynomials, factorization helps in simplifying expressions and solving equations.
To factor a polynomial means to find a set of polynomial expressions of lower degree that multiply to give the original polynomial.

• For example, the polynomial \( x^2 - 5x + 6 \) can be factored into \((x-2)(x-3)\), both of which are factors of the original polynomial.
• Factorization is beneficial because it converts complex polynomial equations into simpler, solvable forms.
• The Factor Theorem links closely with factorization by identifying potential factors of a polynomial through substitution and evaluation techniques.

For the factor theorem to indicate that \( x-c \) is a factor, \( f(c) \) must equal zero. If \( f(c) ot= 0 \), as we saw in the exercise, \( x-c \) is not a factor, making factorization crucial for understanding polynomial division and zero determination.

Real Zeros

Real zeros, or roots, of a polynomial are the \( x \)-values at which the polynomial function evaluates to zero. These are critically important in understanding the behavior of polynomial graphs.
Real zeros are where the graph of the polynomial touches or crosses the x-axis. Finding these zeros gives insights into the solutions of polynomial equations.

• If a polynomial \( f(x) \) has a real zero at \( x = c \), then \( f(c) = 0 \).
• The Factor Theorem provides that if \( f(c) = 0 \), then \( x-c \) is a factor of the polynomial. Hence, identifying real zeros helps in the factorization process.
• The number of real zeros, up to the degree of the polynomial, indicates how many times the polynomial will intersect the x-axis.

Understanding real zeros is essential for solving polynomial equations and for real-life applications where these zeros can represent important quantities, like profit maximization or finding equilibrium points in economics.