Problem 43 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The value of \(P(-2)\) is \(-7\).

Step 1: Set Up Synthetic Division

Place the value of \(c = -2\) to the left, and the coefficients of \(P(x) = x^3 + 2x^2 + 0x - 7\) in a row. The coefficients are \(1, 2, 0, -7\). We include the \(0\) because there is no \(x\) term in the polynomial.

Step 2: Begin the Synthetic Division Process

Bring down the leading coefficient, \(1\), entirely below the line. This is the start of your synthetic division process.

Step 3: Multiply and Add

Multiply \(-2\) by the number just written below the line, which is \(1\), and write the result, \(-2\), below the next coefficient, \(2\). Now, add \(2\) and \(-2\) to get \(0\), and write \(0\) below the line.

Step 4: Continue the Process

Repeat the previous step. Multiply \(-2\) by \(0\) (the result from step 3) to get \(0\), and add it to the next coefficient, \(0\), resulting in \(0\). Write this result below the line.

Step 5: Final Multiplication and Addition

Again, multiply \(-2\) by \(0\) (the latest result), getting \(0\), and add to \(-7\) (the last constant in the polynomial), which results in \(-7\). Write \(-7\) below the line.

Step 6: Interpret the Remainder

The final value written below the line, \(-7\), is the remainder. According to the Remainder Theorem, this is the value of \(P(-2)\).

Key Concepts

Remainder Theorem

The Remainder Theorem is a nifty tool in algebra, especially when working with polynomials. Its primary function is to simplify the evaluation of polynomials at a specific point, known as the value of \(c\). The theorem states that if you divide a polynomial \(P(x)\) by \(x - c\), the remainder you obtain is exactly \(P(c)\). This means you don’t have to plug \(c\) directly into the polynomial and calculate everything manually.
Instead, use synthetic division or long division techniques to quickly find the remainder, which provides the same result. This can save a lot of time, especially with polynomials of high degree or with complex coefficients.

• Use this theorem to check roots: if the remainder is 0 when you divide by \(x - c\), then \(c\) is a root of the polynomial.
• It helps in sketching polynomial graphs by identifying points that the polynomial will pass through.

In practice, as shown in the exercise, once you finish the synthetic division, the last value you land on is the remainder. In this case, for \(P(-2)\), it was \(-7\), confirming the Remainder Theorem's results.

Polynomial Evaluation

Polynomial Evaluation is the process of determining the value of a polynomial at a given point. Normally, you substitute the value of the variable and perform the necessary arithmetic operations. However, this can sometimes be cumbersome, especially with higher degree polynomials or large coefficients.
Instead, synthetic division offers a quicker method. By setting up the division with the provided value of \(c\) and the coefficients of the polynomial, you can determine the polynomial's value at \(c\) without direct substitution.

• This involves adding and multiplying steps that move left across the polynomial.
• Saves time and reduces chances of errors from manual calculations.

In the provided exercise, evaluating \(P(-2)\) using synthetic division resulted in a remainder of \(-7\), showing this efficient process in action.

Polynomial Coefficients

When dealing with polynomials, understanding coefficients is essential. These are the numerical factors that multiply each term of the polynomial. In the polynomial \(P(x) = x^3 + 2x^2 + 0x - 7\), the coefficients are \(1, 2, 0, -7\).
They tell us the weight or significance of each corresponding term in the polynomial and are essential when performing methods like synthetic division. Coefficients should always be aligned correctly, and if any term is missing, like the \(x\) term here, you include a 0 to represent it, ensuring every potential power of \(x\) is accounted for.

• The leading coefficient is the first in the list, here it's 1 for \(x^3\).
• Even zero coefficients play a role in properly framing the equation for division tasks.

In synthetic division, these coefficients help create a straightforward pattern for operations, simplifying what could otherwise be an overwhelming polynomial evaluation.