Problem 36 - Precalculus with Calculus Previews

Question

Use synthetic division and the Remainder Theorem to find \(f(c)\) for the given value of c. $$ f(x)=3 x^{4}-5 x^{2}+27 ; c=\frac{1}{2} $$

Step-by-Step Solution

Answer

\(f\left(\frac{1}{2}\right) = \frac{435}{16}\).

Step 1: Set up the synthetic division

Since we are given that \(c = \frac{1}{2}\), the synthetic division will use \(\frac{1}{2}\) as the divisor. Set up the division by writing down the coefficients of the polynomial \(f(x) = 3x^4 + 0x^3 - 5x^2 + 0x + 27\). The coefficients are \([3, 0, -5, 0, 27]\). This is because there are no \(x^3\) and \(x\) terms explicitly in the polynomial, so their coefficients are zero.

Step 2: Perform synthetic division

Bring down the leading coefficient, \(3\), as it is. Multiply \(3\) by \(\frac{1}{2}\) to get \(\frac{3}{2}\), and write it under the next coefficient (0). Add \(0 + \frac{3}{2} = \frac{3}{2}\). Multiply \(\frac{3}{2}\) by \(\frac{1}{2}\) to get \(\frac{3}{4}\), and write it under the next coefficient \(-5\). Then, \(-5 + \frac{3}{4} = -\frac{17}{4}\). Continue this process for all coefficients.

Step 3: Calculate remainder

Continue with synthetic division: Multiply \(-\frac{17}{4}\) by \(\frac{1}{2}\) to get \(-\frac{17}{8}\), add \(0 + (-\frac{17}{8}) = -\frac{17}{8}\). Now, multiply \(-\frac{17}{8}\) by \(\frac{1}{2}\) to get \(-\frac{17}{16}\), and add \(27 + (-\frac{17}{16})\) to get the remainder, \(\frac{435}{16}\).

Step 4: Apply the Remainder Theorem

According to the Remainder Theorem, the remainder we get from synthetic division is equal to \(f(c)\). Since we found the remainder to be \(\frac{435}{16}\), it follows that \(f\left(\frac{1}{2}\right) = \frac{435}{16}\).

Key Concepts

Remainder Theorem

The Remainder Theorem is a useful shortcut for evaluating polynomial expressions. Its main idea is that when you divide a polynomial \(f(x)\) by a linear divisor of the form \(x - c\), the remainder of this division will be equal to \(f(c)\). Instead of plugging in the value and calculating directly, you perform a division to quickly find the result.

• If \(f(x)\) is divided by \(x - c\), then \(f(c)\) is the remainder.
• This is helpful in checking if a value is a root of a polynomial without long calculations.
• If the remainder is zero, this means \(c\) is a root of the polynomial.

Practically, this theorem speeds up the process of polynomial evaluation and simplifies tasks that would otherwise require complex arithmetic.

Polynomial Division

Polynomial division refers to dividing a polynomial by another, which can be either a simpler polynomial or a constant. A specific technique used for this purpose is synthetic division, an efficient method especially when dividing by linear expressions like \(x - c\).

To perform synthetic division, you take only the coefficients of the polynomial and apply a series of simple operations:

• Arrange the polynomial in descending order of power and write down the coefficients.
• Set the divisor to be the root value \(c\) from \(x - c\).
• Perform a series of multiply-and-add operations among these coefficients starting with the leading one.

This method reduces computation time drastically compared to long division, making it a preferred choice when a polynomial divisor is linear. It’s a more straightforward approach and often used in conjunction with the Remainder Theorem.

Polynomial Functions

Polynomial functions are mathematical expressions formed by variables raised to whole number powers, combined with coefficients. They have a wide range of applications in different fields such as engineering, economics, and natural sciences.

A polynomial function is described by the equation \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \,\ldots\, + a_1 x + a_0\).

\(a_n, a_{n-1},\, ...,\, a_0\) are called the coefficients, and \(n\) is the degree of the polynomial.

These functions have smooth, continuous curves and the highest power of the variable determines the shape and behavior of the curve.
In practical terms:

• The degree of the polynomial indicates the maximum number of roots it can have.
• They help in modeling real-world situations where relationships are non-linear but predictable.

Understanding polynomial functions is essential for solving various algebraic problems, including those involving polynomial division and the Remainder Theorem.