Problem 11 - Algebra for College Students

Question

For Problems \(11-20\), find \(f(c)\) either by using synthetic division and the remainder theorem or by evaluating \(f(c)\) directly. $$ f(x)=6 x^{5}-3 x^{3}+2 \text { and } c=-1 $$

Step-by-Step Solution

Answer

The value of \( f(-1) \) is \(-1\).

Step 1: Set up the function and the constant

We have the polynomial function \( f(x) = 6x^5 - 3x^3 + 2 \). The constant provided is \( c = -1 \). We are tasked with finding \( f(-1) \).

Step 2: Evaluate the function directly

Substitute \( x = -1 \) into the polynomial function \( f(x) \). We compute \( f(-1) = 6(-1)^5 - 3(-1)^3 + 2 \).

Step 3: Simplify each term

Calculate each component separately: - \((-1)^5 = -1\), so \(6(-1)^5 = 6(-1) = -6\).- \((-1)^3 = -1\), so \(-3(-1)^3 = -3(-1) = 3\).- The constant term remains \(+2\).

Step 4: Sum the terms

Add the calculated values from the previous step: \[-6 + 3 + 2 = -1\].

Step 5: Conclusion: Determine the value of \( f(c) \)

Thus, the value of the polynomial \( f(x) \) when \( x = -1 \) is \( f(-1) = -1 \).

Key Concepts

Synthetic Division

Synthetic division is a method used to divide polynomials quickly. It's an efficient shortcut over long division, especially when dividing by a linear factor like \(x - c\). This process simplifies complex calculations and is particularly useful in certain polynomial function scenarios.
To use synthetic division:

• Identify the coefficients of the polynomial and the constant, \(c\).
• Arrange the coefficients in descending order of power, filling in zeros for any missing terms.
• Carry down the leading coefficient to start your calculations.
• Multiply \(c\) by the number just carried down, add the result to the next coefficient, and repeat these steps through all coefficients.

This process helps determine the quotient and remainder of the division. The remainder gives us insights about the function's value at a particular constant as used in the remainder theorem.
Generally simpler than direct computation, synthetic division is an essential tool in a mathematician’s toolkit.

Remainder Theorem

The remainder theorem provides a quick way to evaluate the value of a polynomial function at a given point. It states that the remainder of the division of a polynomial \(f(x)\) by a linear divisor \(x - c\) is equivalent to \(f(c)\). This means if you divide the polynomial by \(x - c\) using methods like synthetic division, the remainder you get is actually \(f(c)\).
This is incredibly useful for checking solutions, like in the given exercise. If you use synthetic division on the polynomial \(f(x) = 6x^5 - 3x^3 + 2\) with \(c = -1\), the remainder will be \(-1\), indicating that \(f(-1) = -1\).
The remainder theorem thus not only aids in polynomial function evaluations but also in understanding polynomial function behaviors and properties quickly. Therefore, it's a foundational concept in algebra and calculus.

Polynomial Functions

Polynomial functions are algebraic expressions involving a sum of powers of variables with coefficients. In simpler terms, they're expressions like \(ax^n + bx^{n-1} + ... + z\). These functions are fundamental in mathematics due to their versatility in modeling a variety of natural phenomena.
Key aspects of polynomial functions include:

• Degree: Determined by the highest power of the variable.
• Coefficients: The numerical factors of each term.
• Constants: The terms without variables.

A polynomial like \(f(x) = 6x^5 - 3x^3 + 2\) is determined by its structure and the operations possible on it, such as addition, subtraction, and multiplication. Evaluating these functions can be done in many ways, such as direct substitution or using special methods like synthetic division and the remainder theorem.
Understanding polynomial functions is crucial as they form the basis for more advanced algebraic concepts, helping students navigate through mathematical problems effectively.