Problem 34 - A Graphical Approach to College Algebra

Question

Use synthetic division to find \(P(k)\). $$k=-2 ; \quad P(x)=x^{2}+5 x+6$$

Step-by-Step Solution

Answer

The remainder is 0, so \( P(-2) = 0 \).

Step 1: Set up the Synthetic Division

First, we set up the synthetic division. Write down the value of \( k = -2 \) on the left. List the coefficients of the polynomial \( P(x) = x^2 + 5x + 6 \) to the right: 1, 5, and 6.

Step 2: Bring Down the Leading Coefficient

Bring down the leading coefficient, which is 1, to the bottom row. This is the starter for the calculations in synthetic division.

Step 3: Multiply and Add

Multiply the value brought down (1) by \( k = -2 \). The result is \( -2 \). Add this to the next coefficient (5), resulting in \( 3 \). Write this result in the row below.

Step 4: Multiply Again

Now, take the result from the previous step (3) and multiply it by \( k = -2 \). The result is \( -6 \). Add this to the next coefficient (6) to get \( 0 \).

Step 5: State the Remainder

The remainder is the last number in the bottom row after finishing the synthetic division steps, which is \( 0 \). Therefore, \( P(-2) = 0 \).

Key Concepts

Polynomial Evaluation

Evaluating polynomials is an essential skill in algebra, helping determine the value of a polynomial function for a specific input. This process involves substituting a given value for the variable in a polynomial expression and simplifying.
For example, when evaluating the polynomial \( P(x) = x^2 + 5x + 6 \) at \( x = -2 \), we substitute \( -2 \) into the polynomial. This substitution turns the expression into \( P(-2) = (-2)^2 + 5(-2) + 6 \).

• Calculate \( (-2)^2 = 4 \).
• Compute \( 5(-2) = -10 \).
• Add the constants: \( 4 - 10 + 6 \).

Combining these, we find that \( P(-2) = 0 \). This means \( P(x) \) evaluates to 0 when \( x = -2 \), showing us how polynomial evaluation can confirm our result using synthetic division.

Remainder Theorem

The Remainder Theorem provides a quick way to evaluate polynomials. It states that the remainder of the division of a polynomial \( P(x) \) by \( (x - k) \) is the value of \( P(k) \). This theorem simplifies finding \( P(k) \) without directly evaluating the polynomial each time.
In the original exercise, by applying synthetic division to \( P(x) = x^2 + 5x + 6 \) with \( k = -2 \), we efficiently find \( P(-2) \).
Using synthetic division:

• The process starts with listing the polynomial's coefficients: \( 1, 5, 6 \).
• Perform divisions as per the synthetic division method.
• The remainder—\( 0 \) here—indicates \( P(-2) = 0 \).

Thus, the Remainder Theorem confirms if you've executed the steps correctly if the remainder equals \( P(k) \), saving time and effort during polynomial evaluation.

Polynomial Coefficients

Polynomial coefficients are crucial components that affect the curve and shape of a polynomial's graph. In the polynomial \( P(x) = x^2 + 5x + 6 \), the coefficients are \( 1, \ 5, \) and \( 6 \). Each plays a specific role:

• The coefficient \( 1 \) is for \( x^2 \), indicating the quadratic term that shapes the parabolic curve.
• The coefficient \( 5 \) is for \( x \), contributing to the slope or tilt of the graph. It affects the direction of the curve's opening.
• The constant \( 6 \) shifts the graph vertically, affecting the position of the polynomial along the y-axis.

In synthetic division, these coefficients determine the calculations and results. As seen in the process, they were essential in each multiplication and addition step, ultimately finding the value of the polynomial at a given point.