Problem 45 - Precalculus Mathematics for Calculus

Question

Use synthetic division and the Remainder Theorem to evaluate \(P(c)\). $$P(x)=5 x^{4}+30 x^{3}-40 x^{2}+36 x+14, \quad c=-7$$

Step-by-Step Solution

Answer

The value of \(P(-7)\) is \(-483\).

Step 1: Set Up Synthetic Division

To use synthetic division, start by setting up the division. In this case, the divisor is \(c = -7\). Write \(-7\) on the left side with space for computation to its right. List the coefficients of the polynomial \(P(x)\) in order: \(5, 30, -40, 36, 14\).

Step 2: Bring Down the Leading Coefficient

Bring down the leading coefficient of the polynomial, which is \(5\), directly to the bottom row.

Step 3: Multiply and Add

Multiply \(-7\) by the number just brought down (\(5\)), which gives \(-35\). Add this to the next coefficient \(30\). The sum is \(-5\). Write this result below the coefficient.

Step 4: Repeat the Process

Repeat the multiply and add process: Multiply \(-7\) by \(-5\) to get \(35\). Add this to the next coefficient \(-40\), resulting in \(-5\). Continue this process: Multiply \(-7\) by \(-5\) to obtain \(35\), then add to \(36\), getting \(71\). Finally, multiply \(-7\) by \(71\), which results in \(-497\), and add to the last coefficient \(14\), arriving at \(-483\).

Step 5: Interpret the Remainder

The final number at the bottom row is the remainder of the division and is the value of \(P(-7)\). Thus, \(P(-7) = -483\), confirming the value of the polynomial at \(x = -7\) using the Remainder Theorem.

Key Concepts

Remainder Theorem

The Remainder Theorem is a powerful tool in algebra. It connects polynomial division with evaluation of polynomials. Specifically, the theorem states that when a polynomial \( P(x) \) is divided by \( x-c \), the remainder is \( P(c) \). This means, to find the value of a polynomial at a particular point, you can simply use division to determine the remainder.

In our case, dividing \( P(x) = 5x^4 + 30x^3 - 40x^2 + 36x + 14 \) by \( x+7 \) (because \( c = -7 \)), we see that the remainder equals \( P(-7) \). Hence, using synthetic division and finding a remainder of \(-483\) confirms that \( P(-7) = -483 \).

The benefit of the Remainder Theorem lies in simplifying the process. It removes the need to perform full polynomial division or plug the number directly into the polynomial and calculate, thereby reducing error and time spent on computation.

Polynomial Evaluation

Evaluating a polynomial means finding its value at a specific point. You substitute the point into the polynomial equation to determine this. However, when dealing with higher-degree polynomials, substituting directly can be tedious.

Here comes the magic of synthetic division, which provides a much faster alternative. Using this approach, we systematically divide the polynomial coefficients by the value derived from \( c \) (our specific point). This technique eased evaluating \( P(-7) \) directly without cumbersome calculations.

Hence, polynomial evaluation becomes more efficient and easier using synthetic division, especially when working with larger polynomials. The result is not only quick but also reliable.

Coefficients

Coefficients are the numerical factors in polynomial terms. For instance, in the polynomial \( P(x) = 5x^4 + 30x^3 - 40x^2 + 36x + 14 \), the coefficients are \( 5, 30, -40, 36, \) and \( 14 \).

In synthetic division, these coefficients play a crucial role. They are listed in sequence and used to perform arithmetic operations that yield results like remainders and other terms.

Understanding the placement and manipulation of coefficients assists in grasping the results of polynomial division. The selected coefficients from each term of \( P(x) \) provide not only a roadmap of the polynomial's structure but also unlock quick solutions through processes like synthetic division. This understanding helps students handle polynomials more effectively, whether in dividing or evaluating them.