Problem 49 - A Graphical Approach to College Algebra

Question

Use synthetic division to determine whether the given number is a zero of the polynomial. $$-0.5 ; \quad P(x)=4 x^{3}+12 x^{2}+7 x+1$$

Step-by-Step Solution

Answer

Yes, -0.5 is a zero of the polynomial.

Step 1: Understand the Problem

We need to determine if \(-0.5\) is a zero of the polynomial \(P(x) = 4x^3 + 12x^2 + 7x + 1\) using synthetic division. A number \(c\) is a zero of a polynomial \(P(x)\) if \(P(c) = 0\). This means we perform synthetic division of the polynomial by \(x + 0.5\) and check if the remainder is zero.

Step 2: Set up Synthetic Division

Set up the synthetic division by writing \(-0.5\) to the left. Write the coefficients of the polynomial \(4, 12, 7, 1\) to the right. The setup looks like this:\[\begin{array}{r|rrrr}-0.5 & 4 & 12 & 7 & 1 \ & & & & \\end{array}\]

Step 3: Perform Synthetic Division

1. Bring down the leading coefficient, 4.2. Multiply \(-0.5\) by 4 and write the result under the second coefficient: \(-0.5 \times 4 = -2\).3. Add to the next coefficient: \(12 + (-2) = 10\).4. Multiply \(-0.5\) by 10: \(-0.5 \times 10 = -5\).5. Add to the next coefficient: \(7 + (-5) = 2\).6. Multiply \(-0.5\) by 2: \(-0.5 \times 2 = -1\).7. Add to the final coefficient: \(1 + (-1) = 0\).

Step 4: Analyze the Remainder

The remainder from the synthetic division is 0. This indicates that \(-0.5\) is indeed a zero of the polynomial since the remainder when divided by \(x + 0.5\) is zero.

Key Concepts

Polynomial Division

When dealing with complex polynomial expressions, dividing them using traditional methods can be quite cumbersome and time-consuming. This is where the beauty of polynomial division comes into play. Synthetic division is a streamlined method specifically designed for dividing polynomials, particularly when you're dividing by a linear factor.

• The main advantage of synthetic division over long division is its simplicity and speed. This method can quickly verify whether a number is a zero of the polynomial.
• In synthetic division, you primarily work with the coefficients of the polynomial, simplifying the process greatly.

Unlike long division, which requires aligning terms and dealing with variables, synthetic division reduces the problem to simple arithmetic with numbers. This makes it particularly useful in algebraic problems involving polynomial functions. If the remainder from synthetic division is zero, it indicates that the divisor is indeed a factor of the polynomial, affirming that the number in question is a zero of the polynomial function.

Zeros of Polynomials

Finding the zeros of a polynomial involves determining the x-values where the polynomial equals zero. These values are crucial as they represent the points where the graph of the polynomial crosses or touches the x-axis. Understanding this helps in graphing polynomial functions and analyzing their behavior. Here’s how synthetic division assists in determining zeros:

• During the division process, if the remainder is zero, the divisor (or the value set for division) is a zero of the polynomial.
• Zeros correspond to solutions of the polynomial equation and are often called roots.

Identifying zeros provides insight into the factorization of the polynomial. In addition to confirming when a particular number is a zero, synthetic division helps break down the polynomial into simpler components, which could be analyzed further for more zeros. Thus, synthetic division is an effective tool for zero-finding in polynomial equations.

Polynomial Functions

Polynomial functions form the backbone of algebra and calculus, representing curves and continuous processes. They are defined as expressions consisting of variables and coefficients, involving terms in the form of powers of a variable. Let's understand a bit more about polynomial functions:

• Polynomial functions can have degrees, which is the highest power of the variable in the function. The degree dictates the complexity and the number of turns in the graph.
• These functions can model a wide array of real-world phenomena, making them central to mathematical modeling.

Polynomials also have unique properties, such as end behavior and symmetry, which help in predicting their graphs. Understanding how these functions behave in different intervals helps students solve complex equations and inequalities easily. Polynomial functions can be broken down into factors (like using synthetic division), simplifying the analysis and graphing of the entire function.