Problem 50 - Precalculus Mathematics for Calculus

Question

Use synthetic division and the Remainder Theorem to evaluate \(P(c)\). $$P(x)=x^{3}-x+1, \quad c=\frac{1}{4}$$

Step-by-Step Solution

Answer

The value of \(P(\frac{1}{4})\) is \(\frac{49}{64}\).

Step 1: Set Up the Synthetic Division

To begin synthetic division, we need to organize the coefficients of the polynomial \(P(x) = x^3 - x + 1\). First, write the coefficients in a row. Since there is no \(x^2\) term, its coefficient is 0:1 (for \(x^3\)), 0 (for \(x^2\)), -1 (for \(x\)), 1 (constant term).Next, write \(c = \frac{1}{4}\) on the left, outside the row of coefficients.

Step 2: Perform the Synthetic Division

Bring down the first coefficient (1) as it is. Multiply \(\frac{1}{4}\) (the number outside) by this 1 and write the result (\(\frac{1}{4}\)) under the next coefficient (0). Add this result to 0, getting \(\frac{1}{4}\). Repeat the process: Multiply \(\frac{1}{4}\) by \(\frac{1}{4}\) to get \(\frac{1}{16}\), write under -1, and add to get \(-\frac{16}{16} + \frac{1}{16} = -\frac{15}{16}\). Finally, multiply \(\frac{1}{4}\) by \(-\frac{15}{16}\) to get \(-\frac{15}{64}\), and add it to 1 to get \(\frac{49}{64}\).

Step 3: Identify the Remainder

The last number obtained from the synthetic division (\(\frac{49}{64}\)) is the remainder. According to the Remainder Theorem, the remainder when dividing \(P(x)\) by \(x - c\) is \(P(c)\). Thus, \(P(\frac{1}{4}) = \frac{49}{64}\).

Key Concepts

Polynomial Evaluation

Evaluating polynomials is a fundamental skill in algebra which often involves substituting a specific value into a polynomial function. In general, if you have a polynomial \(P(x)\) and you wish to evaluate it at \(x=c\), like in our expression, you would simply replace all instances of \(x\) with \(c\). For instance, substituting \(x=\frac{1}{4}\) into the polynomial \(P(x) = x^3 - x + 1\) gives \(P\left(\frac{1}{4}\right)\). In some cases, performing this substitution directly can be cumbersome, particularly for higher-degree polynomials. Thus, techniques like synthetic division become highly useful to simplify calculations.

Remainder Theorem

The Remainder Theorem is an essential concept that connects polynomial division with evaluation. It states that for a polynomial \(P(x)\), the remainder of dividing \(P(x)\) by \(x-c\) equals \(P(c)\). This means instead of evaluating \(P(c)\) directly, you can perform polynomial division and obtain the result from the remainder. For our exercise, using synthetic division, we determined that the remainder when dividing \(P(x) = x^3 - x + 1\) by \(x-\frac{1}{4}\) is \(\frac{49}{64}\). This value is precisely \(P\left(\frac{1}{4}\right)\), confirming the Remainder Theorem.

Polynomial Coefficients

Polynomials are expressions that consist of several terms, with each term containing a coefficient and a variable raised to a power. The coefficients dictate the structure of the polynomial and are crucial in synthetic division. In the polynomial \(P(x) = x^3 - x + 1\), the coefficients are:

• 1 for the \(x^3\) term.
• 0 for the missing \(x^2\) term.
• -1 for the \(x\) term.
• 1 for the constant term.

These coefficients are used in synthetic division to facilitate the polynomial evaluation, especially when the polynomial's degree is higher and direct substitution is not straightforward.

Problem Solving Steps

Solving this type of problem systematically involves a few distinct steps, each building upon the last. Let's break it down:

First, prepare for synthetic division by aligning the polynomial coefficients and place the given value \(c\) outside the array. With \(P(x) = x^3 - x + 1\), our coefficients are \,1, 0, -1, 1\. We place \(c = \frac{1}{4}\) outside.

Next, start the division. Bring down the leading coefficient (1) directly, multiply \(c\) by this and add downwards for each step. Continue this for each coefficient:

• Multiply \(\frac{1}{4} \times 1 = \frac{1}{4}\), add to 0 to get \(\frac{1}{4}\).
• Multiply \(\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}\), add to -1 to get \(-\frac{15}{16}\).
• Multiply \(\frac{1}{4} \times -\frac{15}{16} = -\frac{15}{64}\), add to 1 to get \(\frac{49}{64}\).

Finally, interpret the result. The last number, \(\frac{49}{64}\), is the remainder, and by the Remainder Theorem, this is our evaluation: \(P\left(\frac{1}{4}\right) = \frac{49}{64}\). This approach not only finds the polynomial's value at \(c\) but also deepens your understanding of algebraic structures.