Problem 35 - A Graphical Approach to Precalculus with Limits

Question

Use synthetic substitution to find \(P(k).\) $$k=-2 ; \quad P(x)=5 x^{3}+2 x^{2}-x+5$$

Step-by-Step Solution

Answer

\(P(-2) = -25\)

Step 1: Set Up the Synthetic Substitution Table

To use synthetic substitution, we first write down the coefficients of the polynomial \(P(x) = 5x^3 + 2x^2 - x + 5\). So, the coefficients are \(5, 2, -1,\) and \(5\). We align them in order for easy manipulation.

Step 2: Bring Down the Leading Coefficient

Bring down the first coefficient, which is \(5\), to start the process. This serves as the first entry in the row where we will compute the remaining values.

Step 3: Multiply and Add

Multiply the number beneath the horizontal line by \(k = -2\) (i.e., \(5 \times -2 = -10\)) and write the result beneath the next coefficient. Then add this result \(-10\) to the next coefficient \(2\), getting \(2 + (-10) = -8\).

Step 4: Repeat Multiply and Add

Repeat the same process: Multiply the result from the previous addition (which is \(-8\)) by \(-2\) to get \(16\). Add this \(16\) to the next coefficient \(-1\), resulting in \(-1 + 16 = 15\).

Step 5: One More Multiply and Add

Multiply the last addition result (\(15\)) by \(-2\), getting \(-30\). Add this \(-30\) to the last coefficient \(5\), yielding \(5 + (-30) = -25\).

Step 6: Identify the Remainder

The value from the last addition (\(-25\)) is the remainder of the synthetic substitution process. This is the value of \(P(-2)\).

Key Concepts

Polynomial Evaluation

Polynomial evaluation is an essential concept that involves calculating the value of a polynomial for a specific variable value. This is done by substituting the given number into the polynomial and performing the arithmetic necessary to find the result. For instance, if you want to evaluate the polynomial \(P(x) = 5x^3 + 2x^2 - x + 5\) at \(x = -2\), you would replace each \(x\) in the polynomial with \(-2\). Then, you compute: - \(5(-2)^3 + 2(-2)^2 - (-2) + 5\) - This leads to \(5(-8) + 2(4) + 2 + 5\) - Resulting in \(-40 + 8 + 2 + 5\) - Which simplifies to \(-25\).This process systematically breaks down the polynomial equation, making it manageable to handle even higher degrees polynomials.

Remainder Theorem

The Remainder Theorem provides a handy shortcut in polynomial math. It states that when a polynomial \(P(x)\) is divided by \(x - k\), the remainder of this division is \(P(k)\), which simplifies the process of evaluating polynomials.So, let's apply this theorem to our example. Given \(P(x) = 5x^3 + 2x^2 - x + 5\) and \(k = -2\), if you were to perform the division \(P(x) \div (x + 2)\), the remainder will be \(P(-2)\).Instead of performing tedious polynomial long division, you can use the Remainder Theorem to immediately know that when you substitute \(-2\) into the polynomial, yielding \(-25\), that \(-25\) is indeed the remainder.

Synthetic Division

Synthetic division streamlines the process of dividing polynomials when the divisor is in the form \(x - k\). This method is less cumbersome than traditional long division, especially for higher degree polynomials, and can also be used to evaluate polynomials quickly through synthetic substitution.Here’s how synthetic division works:

• Write down the coefficients of the polynomial \(5, 2, -1,\) and \(5\).
• Use \(k = -2\) (as in our exercise) for synthetic substitution.
• Bring down the first coefficient, \(5\), unchanged.
• Multiply \(5\) by \(-2\), and place this result under the second coefficient.
• Continue to multiply and add as described in the original exercise.

This technique not only helps in simplifying polynomial division but also proves useful in polynomial evaluation and finding roots of polynomial equations. Through steps like these, it offers both a quick solution and better understanding.