Problem 45 - Algebra and Trigonometry

Question

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 6479.

Step 1: Set Up the Synthetic Division

To use synthetic division, arrange the coefficients of the polynomial in order. For \(P(x) = x^7 + 0x^6 + 0x^5 + 0x^4 + 0x^3 - 3x^2 + 0x - 1\), the coefficients are \([1, 0, 0, 0, 0, -3, 0, -1]\). Use \(c = 3\) for the synthetic division.

Step 2: Synthetic Division Process

Write the number \(3\) on the left. Bring down the leading coefficient \(1\) to the bottom row. Multiply \(3\) by \(1\), write the result under the next coefficient. Add it to the coefficient above. Continue this for all coefficients.

Step 3: Calculate Synthetic Division Table

Perform the calculations step-by-step:\(\begin{array}{ccccccccc}3 & | & 1 & 0 & 0 & 0 & 0 & -3 & 0 & -1 \ & & & 3 & 9 & 27 & 81 & 243 & 720 & 2160 & 6480 \\hline & & 1 & 3 & 9 & 27 & 81 & 240 & 720 & 2160 & 6479 \\end{array}\)- Write down the remaining numbers: The result of division leaves a remainder of 6479.

Step 4: Use Remainder Theorem

According to the Remainder Theorem, the remainder of the division of \(P(x)\) by \(x - c\) is \(P(c)\). Thus, \(P(3) = 6479\).

Key Concepts

remainder theorem

The Remainder Theorem is a powerful tool that connects directly to the concept of polynomial division. It states that if you divide a polynomial \( P(x) \) by a linear divisor \( x-c \), then the remainder of this division is exactly \( P(c) \). Essentially, this theorem provides a quick and easy way to evaluate polynomials at specific points without needing to perform the entire polynomial division.Suppose you have a polynomial and you want to find the value of the function at \( x = c \). You don't need to substitute \( c \) into the polynomial directly. Instead, you can perform the division of \( P(x) \) by \( x-c \) and simply look at the remainder. This remainder is equal to \( P(c) \).For example, in the given polynomial \( P(x) = x^7 - 3x^2 - 1 \) and \( c = 3 \), after performing the synthetic division, the remainder we find is 6479. According to the Remainder Theorem, this remainder is the value of \( P(3) \). This short-cuts the rneed for direct evaluation and simplifies the whole process.

polynomial evaluation

Polynomial evaluation involves finding the value of a polynomial function at a given point. Instead of calculating each term of the polynomial separately, there are more efficient methods like synthetic division that can be used.For the polynomial \( P(x) = x^7 - 3x^2 - 1 \), evaluating \( P(3) \) means we are interested in the function's output when \( x = 3 \). The traditional method would involve plugging 3 into each term of the polynomial, calculating powers like \( 3^7 \), and then combining all these numbers — which can be cumbersome and time-consuming for large polynomials.Using synthetic division streamlines this process. By aligning the coefficients of the polynomial and employing the Remainder Theorem, we can evaluate the polynomial efficiently. In this instance, the synthetic division process quickly reveals that \( P(3) = 6479 \), allowing us to evaluate the polynomial accurately with minimal arithmetic.

polynomial division

Polynomial division is a fundamental skill in algebra that is similar to long division but adapted for polynomials. It helps simplify complex polynomials and can be performed using methods like long division or synthetic division.Synthetic division is especially useful when dividing a polynomial by a linear factor of the form \( x-c \). In this process, only the coefficients of the polynomial are involved, which simplifies calculations. Synthetic division is faster and more straightforward than traditional long division when the divisor is linear.Here’s a brief recap of how the synthetic division was applied to our problem:

• Start with the coefficients of the polynomial: \([1, 0, 0, 0, 0, -3, 0, -1]\).
• Use the value \( c = 3 \) for division.
• Bring down the leading coefficient and continue the process of multiplying and adding down the columns.
• The final number calculated is the remainder, giving the result of \( P(3) = 6479 \).

This method not only provides division results but also aligns seamlessly with the Remainder Theorem, providing efficient polynomial evaluation.