Problem 46 - A Graphical Approach to College Algebra

Question

Use synthetic division to determine whether the given number is a zero of the polynomial. $$-1 ; \quad P(x)=x^{2}+4 x-5$$

Step-by-Step Solution

Answer

No, \(-1\) is not a zero of the polynomial.

Step 1: Identify the Coefficients

Write down the coefficients of the polynomial \( P(x) = x^2 + 4x - 5 \). The coefficients are 1, 4, and -5.

Step 2: Set Up Synthetic Division

Write the number you are testing, \(-1\), at the top left corner, and draw a horizontal and vertical line to the right of it to separate it from the coefficients. Write the coefficients in a row to the right of these lines.

Step 3: Bring Down the First Coefficient

Bring down the first coefficient, 1, directly below the line. This acts as the initial number in the synthetic division process.

Step 4: Multiply and Add

Multiply \(-1\) by the number below the line (initially 1), giving \(-1\). Write this result under the next coefficient (4). Add the numbers in the column (4 + (-1)) to get 3, and write this under the line.

Step 5: Repeat Multiply and Add

Multiply \(-1\) by the most recent result below the line (3), giving \(-3\). Write this under the next coefficient (-5). Add the numbers in the column (-5 + (-3)) to get \(-8\), and write this under the line.

Step 6: Interpret the Remainder

Look at the last number written under the line, which is \(-8\). This is the remainder obtained from the synthetic division.

Step 7: Determine if \(-1\) is a Zero

If the remainder is zero, then \(-1\) is a zero of the polynomial. Since the remainder is \(-8\), \(-1\) is not a zero of the polynomial \(x^2 + 4x - 5\).

Key Concepts

Polynomial Division

Polynomial division allows us to divide a polynomial by another polynomial of lower degree, just like how you might divide numbers. This process helps in reducing the complexity of problems and makes computations easier.

• When dividing polynomials, each term of the dividend (the polynomial being divided) is divided by the divider (another polynomial), similar to long division.
• Synthetic division is a shortcut, a simplified form of polynomial division where only coefficients are considered, not the variables.

One of the significant benefits of synthetic division is that it is particularly useful for dividing a polynomial by a linear factor of the form \(x - c\), where \(c\) is a constant, or simply when we're checking if \(c\) is a zero of the polynomial.
This approach saves time and avoids potential errors compared to the standard polynomial division procedure.

Zeros of Polynomial

Zeros of a polynomial are the values for which the whole polynomial equals zero. They are also referred to as roots or solutions of the polynomial.

• To find the zeros, we set the polynomial equal to zero and solve the resulting equation.
• Zeros are important because they help us understand the behavior of a polynomial, such as where it crosses the x-axis on a graph.

In the context of synthetic division, testing whether a number \(c\) is a zero, involves plugging \(c\) into the division process. If the remainder from synthetic division is zero, this indicates that \(c\) is indeed a zero of the polynomial. Otherwise, if the remainder is not zero, as with \(-1\) in our example, then \(c\) is not a zero.

Remainder Theorem

The Remainder Theorem offers a quick insight into checking zeros of a polynomial using synthetic division. It states that the remainder of a polynomial \(P(x)\) divided by a linear divisor \(x - c\) is equal to \(P(c)\).

• Using the synthetic division, we can compute \(P(c)\) without directly substituting \(c\) into the polynomial; instead, you just use the coefficients.
• If \(P(c) = 0\), \(c\) is a zero of the polynomial, confirming the factor \(x - c\) is indeed a divisor of \(P(x)\).

For instance, while testing \(-1\) for zeros using the example polynomial, the non-zero remainder \(-8\) implies \(P(-1) eq 0\), hence \(-1\) is not a factor. This outcome effectively demonstrates the application of the Remainder Theorem.