Problem 12 - Algebra and Trigonometry with Analytic Geometry

Question

$$ \text { Use the remainder theorem to find } f(c) \text {. } $$ $$ f(x)=x^{4}+3 x^{2}-12 ; \quad c=-2 $$

Step-by-Step Solution

Answer

The value of \( f(-2) \) is 16.

Step 1: Understand the Polynomial Function

The given polynomial function is \( f(x) = x^4 + 3x^2 - 12 \). We are supposed to find the value \( f(c) \) where \( c = -2 \).

Step 2: Apply the Remainder Theorem

The remainder theorem states that the remainder of the division of a polynomial by \( x - c \) is \( f(c) \). This means that to find \( f(-2) \), we need to evaluate the polynomial at \( x = -2 \).

Step 3: Substitute and Compute \( f(-2) \)

Substitute \(-2\) into the function: \[f(-2) = (-2)^4 + 3(-2)^2 - 12 \]Calculate each term:- \((-2)^4 = 16\)- \(3(-2)^2 = 3 \times 4 = 12\)Subtracting \(12\) gives us:\[f(-2) = 16 + 12 - 12 = 16\]

Step 4: Conclude the Calculation

After evaluating, we find that \( f(-2) = 16 \). This result is the remainder when the polynomial is divided by \( x + 2 \).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions made up of variables, coefficients, and constant terms combined using addition, subtraction, multiplication, and non-negative integer exponents. For example, the polynomial function given in our problem is \( f(x) = x^4 + 3x^2 - 12 \). This function consists of four terms: the highest degree term \( x^4 \), the middle term \( 3x^2 \), and the constant term \( -12 \).

Polynomials can be classified by degree (the highest power of the variable) and the number of terms. This function is a degree 4 polynomial due to the variable raised to the power of 4. Understanding the structure of polynomial functions helps in solving them using various algebraic tools like factoring or evaluating at specific values.

These functions are central to algebra because they form the foundation for more complex mathematics. They appear in equations, graphs, and real-life models where predicting outcomes or reactions is needed.

Evaluating Polynomials

Evaluating a polynomial means finding its value at a certain point, which in this problem zone is determined by the Remainder Theorem. To evaluate \( f(x) \) at \( x = -2 \), we substitute the value into the polynomial equation. This evaluation transforms our mathematical expression into numerical terms.

Here’s how it works step-by-step:

• Replace every occurrence of \( x \) in the polynomial with \( -2 \).
• Compute each term separately: \( (-2)^4 = 16 \) and \( 3(-2)^2 = 12 \).
• Sum these values: \( 16 + 12 \), then subtract any constant term included, which is \( 12 \) in this instance.

Evaluating polynomials can be approached with techniques like this because it simplifies potentially complex algebra into straightforward arithmetic.

Algebraic Substitution

Algebraic substitution is a fundamental technique used often in evaluating polynomials, solving equations, and simplifying expressions. It involves replacing a variable with a given number or another expression. In our exercise, substitution is key for applying the Remainder Theorem to find \( f(-2) \).

To perform substitution accurately:

• Identify the variable(s) that need substitution. In this case, \( x \) is replaced by \( -2 \).
• Ensure each occurrence of the variable throughout the polynomial is correctly substituted.
• Complete the arithmetic operations as per regular order of operations like parentheses and exponents first.

This method not only simplifies solving polynomials but also plays a crucial role in more advanced math, especially in calculus where functions are evaluated for limits, derivatives, and integrals.