Problem 47 - Algebra and Trigonometry

Question

\(39-51\) . Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) $$ P(x)=x^{7}-3 x^{2}-1, \quad c=3 $$

Step-by-Step Solution

Answer

The value of \(P(3)\) is \(-793\).

Step 1: Understanding Synthetic Division Set-Up

To use synthetic division, we first need to set up our polynomial using the coefficients of each term in descending order of degree. The polynomial given is \(P(x) = x^7 - 3x^2 - 1\), which can be written as \([1, 0, 0, 0, 0, -3, 0, -1]\). The value for \(c\) is given as 3.

Step 2: Setting Up the Synthetic Division Table

Write the value of \(c = 3\) on the left and list the coefficients of the polynomial on the right: \([1, 0, 0, 0, 0, -3, 0, -1]\). Draw a horizontal line under the list of coefficients.

Step 3: Performing the Division

1. Bring down the first coefficient (1) directly below the line.2. Multiply it by \(c = 3\) and write the result underneath the second coefficient (0).3. Write the sum of the second coefficient and the result just below the line.4. Repeat these steps (multiplication and addition) for all coefficients.

Step 4: Completing Synthetic Division

After completing the synthetic division for all terms, the number at the bottom right is the remainder of the division of the polynomial by \(x-c\). In this case, continue the procedure until the end: - The sequence of adding and multiplying gives you a final remainder.

Step 5: Interpreting the Remainder

The remainder obtained from the synthetic division is the value of \(P(3)\) due to the Remainder Theorem. After performing synthetic division, the remainder obtained is \(-793\).

Key Concepts

Remainder Theorem

The Remainder Theorem is a useful concept in algebra that tells you what the remainder of the division of a polynomial by a linear divisor of the form \(x - c\) will be. Specifically, it states that for a given polynomial \(P(x)\), if you divide it by \(x - c\), the remainder of this division is simply \(P(c)\).
This theorem simplifies the process of finding the value of the polynomial at a particular value of \(x\) without performing a full polynomial division.
In our exercise, we are asked to use synthetic division to evaluate \(P(x)\) at \(x = 3\), which directly leverages the Remainder Theorem. We perform synthetic division and the remainder we obtain, which is \(-793\), is actually \(P(3)\).
This theorem simplifies the work considerably and provides an efficient way to check the value of a polynomial at specific points.

Polynomial Division

Polynomial division is a process similar to long division with numbers but applied to polynomials. It allows us to divide one polynomial by another, typically of lower degree, to produce a quotient and a remainder.
Synthetic division is a streamlined method of dividing a polynomial by a linear polynomial of the form \(x - c\).
Here are the steps generally involved in synthetic division:

• Write down the coefficients of the polynomial in a row.
• Place the value of \(c\) (from \(x - c\)) to the left of the line.
• Bring down the leading coefficient to start the second row.
• Multiply \(c\) by each number just written above and add it to the next coefficient.

Synthetic division is particularly useful for quick calculations as it avoids polynomial multiplication and subtraction, reducing complex division into simple addition and multiplication. In this exercise, the divisor is \(x - 3\), requiring us to perform these steps with \(c = 3\).

Polynomial Evaluation

Evaluating a polynomial at a specific value is essentially determining the output of the polynomial function for a particular input. Using synthetic division is one method to evaluate, especially when the remainder theorem applies, as we've seen here.
For the polynomial \(P(x) = x^7 - 3x^2 - 1\), evaluating it at \(x = 3\) using synthetic division provides the remainder as the result of \(P(3)\).
This means:

• You set up your division with \(c = 3\).
• Perform synthetic division using the coefficients of \(P(x)\).
• Read the remainder, which represents \(P(3)\).

With this approach, you don't need to manually substitute \(x = 3\) into every term of the polynomial, but can instead perform a more efficient calculation through division. This ability to rapidly evaluate polynomials is valuable for checking roots or assessing polynomial behavior at certain points.