Problem 37 - College Algebra and Calculus: An Applied Approach

Question

Write the function in the form \(f(x)=(x-k) q(x)+r\) for the given value of \(k\), and demonstrate that \(f(k)=r\). $$f(x)=x^{3}+x^{2}-12 x+20, \quad k=3$$

Step-by-Step Solution

Answer

The function is \(f(x) = (x-3)(x^2+4x)+20\). It verifies correctly as \(f(3) = 20\), which gives \(r=20\).

Step 1: Format the Equation for Polynomial Division

Arrange the equation coefficients and constants in the form of polynomial division and write the function, \(f(x)=x^{3}+x^{2}-12x+20\), as (1, 1, -12, 20). These coefficients represent the values for the variables \(x^3\), \(x^2\), \(x^1\), and \(x^0\) respectively. Now, the divisor \(x-k\) is \(x-3\).

Step 2: Perform Synthetic Division

Synthetic division is done by setting up a division tableau. Write down the coefficients (1, 1, -12, 20) in a row and bring the first coefficient down. Multiply this by the divisor (3 in this case) and add to the next coefficient, repeat this process till the end. Thus, the row becomes (1, 4, 0, 20) and the last term is the remainder (r).

Step 3: Formulate \(q(x)\) and \(r\)

The coefficients (1, 4, 0) represent the polynomial \(q(x) = x^{2} + 4x\) and the remainder \(r=20\). Now the function takes this form: \(f(x)=(x-k)q(x)+r=(x-3)(x^2+4x)+20\).

Step 4: Verify Solution

To verify our work, plug \(k=3\) into the equation \(f(x)\). Doing this, the equation becomes \(f(3)=(3-3)(3^2+4*3)+20=0+20=20\). As \(f(k)=r\) and \(r=20\), this confirms our earlier finding.

Key Concepts

Polynomial Function

A polynomial function is an expression consisting of variables, coefficients, and exponents that are combined using addition, subtraction, and multiplication. It takes the form

• f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0

where each term is made up of the product of a constant and a non-negative integer power of the variable.
The highest power of the variable is known as the degree of the polynomial.
For example, in the polynomial function given by

• f(x) = x^3 + x^2 - 12x + 20

we have:

• Degree: 3 (shown by the term with exponent 3)
• Leading Coefficient: 1 (the coefficient of the highest degree term)
• Constant Term: 20 (the term without any variable)

Understanding polynomial functions is crucial, as they form the basis for solving more complex mathematical computations.
They represent a class of smooth and continuous functions, making them particularly useful in calculus and mathematical modeling.

Remainder Theorem

The Remainder Theorem is a fundamental concept that connects the division of polynomials with evaluating polynomial functions. It states that for a polynomial

• f(x)

divided by a linear divisor of the form

• x - k

the remainder of the division is

• f(k).

This theorem provides a quick and efficient way to find the remainder without performing the full polynomial division process.
To demonstrate and verify the theorem, we followed these steps in the exercise:

• Performed synthetic division of the polynomial f(x) = x^3 + x^2 - 12x + 20 by x - 3. The remainder, according to this process, is found to be 20.
• Substituted the divisor root ( k = 3) into f(x), resulting in f(3) = 20

This confirms the Remainder Theorem, where f(k) = r.

Polynomial Division

Polynomial division is analogous to the long division process taught in elementary mathematics, but it applies to polynomials. It involves dividing a polynomial

• f(x)

by another polynomial, usually a linear term like

• x - k.

In our exercise, this was achieved through synthetic division, a streamlined method ideal for dividing by linear polynomials.
Here’s how synthetic division simplifies the process:

• Setup: Arrange the coefficients of the divisor and dividend.
• Process: Using the root of the divisor, perform iterative steps to reduce the polynomial degree by 1 with each operation.
• Outcome: Quick identification of the quotient polynomial and remainder.

For the polynomial f(x) = x^3 + x^2 - 12x + 20, synthetic division with x - 3 yielded the quotient q(x) = x^2 + 4x and the remainder r = 20.
These results allow us to express f(x) in the form (x-k)q(x) + r, demonstrating the effective application of polynomial division principles.