Problem 4 - Algebra for College Students

Question

For Problems 1-10, find \(f(c)\) by (a) evaluating \(f(c)\) directly, and (b) using synthetic division and the remainder theorem. $$ f(x)=x^{3}+3 x^{2}-4 x-7 \text { and } c=-2 $$

Step-by-Step Solution

Answer

Both methods show that \(f(-2) = 5\).

Step 1: Evaluate f(c) Directly

To find \(f(-2)\), substitute \(x = -2\) into the function \(f(x)\).\[ f(-2) = (-2)^3 + 3(-2)^2 - 4(-2) - 7 \] Calculate each term:- \((-2)^3 = -8\)- \(3(-2)^2 = 3(4) = 12\)- \(-4(-2) = 8\)- Constant \(-7\).Now, sum these values: \[ -8 + 12 + 8 - 7 = 5 \]So, \(f(-2) = 5\).

Step 2: Set Up Synthetic Division

For synthetic division, list the coefficients of \(f(x)\): \(1\) (for \(x^3\)), \(3\) (for \(x^2\)), \(-4\) (for \(x\)), and \(-7\) (constant term). Use \(c = -2\) as the divisor.Set up the synthetic division tableau:-2 | 1 3 -4 -7 |

Step 3: Perform Synthetic Division

Start synthetic division:1. Bring down the leading coefficient (1).2. Multiply by the divisor \(-2\): \[ -2 \times 1 = -2 \] Add to the next coefficient (3): \[ 3 + (-2) = 1 \]3. Multiply \(1\) by \(-2\): \[ -2 \times 1 = -2 \] Add to the next coefficient (-4): \[ -4 + (-2) = -6 \]4. Multiply \(-6\) by \(-2\): \[ -2 \times -6 = 12 \] Add to the next coefficient (-7): \[ -7 + 12 = 5 \]The remainder from synthetic division is 5.

Step 4: Verify Using Remainder Theorem

The Remainder Theorem states that if a polynomial \(f(x)\) is divided by \(x - c\), the remainder is \(f(c)\). Since our remainder from synthetic division is 5, this confirms that \(f(-2) = 5\).

Key Concepts

Remainder Theorem

The Remainder Theorem is a useful tool in algebra for evaluating polynomial functions. It states that if you divide a polynomial function \(f(x)\) by \(x - c\), the remainder of this division will be \(f(c)\). Essentially, this theorem gives us a shortcut. Instead of manually calculating \(f(c)\), you can perform synthetic division and use the remainder to find your answer.

• For example, when you calculate \(f(-2)\) using synthetic division for the polynomial \(f(x) = x^3 + 3x^2 - 4x - 7\), the remainder turns out to be 5. This confirms that \(f(-2) = 5\).

Understanding this theorem simplifies finding values of polynomials at specific points. Thus, you don't just evaluate directly, you can use a more systematic approach.

Polynomial Functions

Polynomial functions are algebraic expressions that consist of variables raised to whole number powers. These expressions can have coefficients and constants, making them flexible for various mathematical applications.

• The general form of a polynomial is \(f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\).
• The degree of a polynomial is the highest power of \(x\) that appears in it. For example, in \(f(x) = x^3 + 3x^2 - 4x - 7\), the degree is 3.
• Understanding the components of polynomial functions is crucial for operations like addition, subtraction, and division.

Polyynomials are foundational in algebra because they describe many relationships in mathematics. This type of function appears in many areas of study, such as calculus, physics, and engineering.

Algebraic Evaluation

Algebraic evaluation involves finding the value of a polynomial expression for a given input. In the original exercise, you were asked to find \(f(-2)\). You do this by substituting \(-2\) into the polynomial and calculating the result.

• For the function \(f(x) = x^3 + 3x^2 - 4x - 7\), you compute \((-2)^3 + 3(-2)^2 - 4(-2) - 7\) to find \(f(-2)\).
• Each step involves basic arithmetic: exponents, multiplication, and addition/subtraction.

Algebraic evaluation is pivotal for solving real-world problems where you need specific values from general formulas. By practicing these steps, you become proficient in evaluating any polynomial function directly.

Synthetic Division Steps

Synthetic division is a streamlined method to divide polynomials. It's faster and more efficient than long division, especially when dealing with linear divisors. Let's break down the steps.

• First, write down the coefficients of the polynomial. For \(f(x) = x^3 + 3x^2 - 4x - 7\), these are 1, 3, -4, and -7.
• Use the value \(c\), in this case \(-2\), as the divisor.
• Set up the division: place \(-2\) on the left and the coefficients on the top row.
• Begin by bringing the leading coefficient straight down. Then multiply it by \(-2\), adding the result to the next coefficient, repeating this until all coefficients are processed.

The final number in your synthetic division tableau is the remainder. If done correctly, this remainder will match \(f(c)\), giving you a quick check of your work. Practicing synthetic division helps build confidence in manipulating polynomials.