Problem 50 - A Graphical Approach to College Algebra

Question

Use synthetic division to determine whether the given number is a zero of the polynomial. $$-0.25 ; \quad P(x)=8 x^{3}+6 x^{2}-3 x-1$$

Step-by-Step Solution

Answer

Yes, \(-0.25\) is a zero of the polynomial.

Step 1: Setup the Synthetic Division Table

To begin the synthetic division process, we need to set up our table. First, we take the polynomial \( P(x) = 8x^3 + 6x^2 - 3x - 1 \) and list out the coefficients: \( 8, 6, -3, -1 \). The divisor here is \(-0.25\) as we are checking if \(-0.25\) is a zero.

Step 2: Bring Down the First Coefficient

The first step in synthetic division is to bring down the first coefficient as is. Thus, the coefficient of \( x^3 \), which is \( 8 \), is brought down to the bottom row.

Step 3: Multiply and Add

Multiply the number \(-0.25\) by the number just brought down (\( 8 \)) and then add it to the next coefficient in the list. \(-0.25 \times 8 = -2\), and add \(-2\) to the coefficient of \(x^2\), which is \(6\), producing a new value of \(4\).

Step 4: Repeat Multiplication and Addition

Repeat this process for the new result that was calculated. Multiply \(-0.25\) by \(4\) (the result from the previous step) and sum it with the next coefficient: \(-0.25 \times 4 = -1\), add \(-1\) to the coefficient of \(x\), \(-3\), to get \(-4\).

Step 5: Final Calculation

Perform the operation one more time for the last coefficient. Multiply \(-0.25\) by \(-4\) and add it to the last coefficient, which is \(-1\): \(-0.25\times(-4) = 1 \), add \(1\) to the constant term \(-1\), resulting in \(0\). This confirms that \(-0.25\) is a zero of the polynomial since the remainder is zero.

Key Concepts

Zero of a Polynomial

Finding a zero of a polynomial is a crucial part of algebra, as it helps identify where the polynomial intersects the x-axis. When we say a number is a zero of a polynomial, it means that substituting this number into the polynomial yields zero. For example, if -0.25 is a zero of \( P(x) = 8x^3 + 6x^2 - 3x - 1 \), it means \( P(-0.25) = 0 \).
Zeros of polynomials are important in understanding the roots and in factorizing polynomials. These zeros are also referred to as the solutions of the polynomial equation \( P(x) = 0 \). Determining these values provides insight into the behavior of the polynomial graph.

• A zero implies the point where the graph of the polynomial touches or crosses the x-axis.
• They are integral to solving polynomial equations and can be real or complex numbers.

Using methods like synthetic division, one can test a given number to see if it's truly a zero. This is accomplished by verifying if substituting the number into the polynomial results in a remainder of zero. This way, you ensure that the polynomial divides exactly without any leftover terms.

Polynomial Roots

The term 'roots of a polynomial' refers to the solutions of the equation \( P(x) = 0 \). Roots have the same meaning as zeros, and they tell us the values for which the polynomial becomes zero. Understanding the roots is essential for graphing and analyzing the polynomial's behavior.
Most polynomials have roots that are either easy to find by looking or require computations, like synthetic division. In the example given, we used synthetic division to confirm that -0.25 is a root. But why is knowing the roots important?

• Roots tell us about the intercepts of the polynomial graph with the x-axis.
• They can inform us on the number of turning points of a polynomial graph, given the degree of the polynomial.
• Roots can be used to reconstruct the polynomial equation by writing it as a product of its factors if all roots are known.

Graphing calculators and software can find roots, but understanding manual methods like synthetic division enrich our comprehension of their significance in polynomials.

Algebraic Techniques

Algebraic techniques are mathematical processes that help solve equations and manipulate expressions. When dealing with polynomials, several techniques can be employed to solve them or simplify them to find roots or zeros.
Synthetic division is one of these powerful algebraic techniques that simplifies dividing polynomials, especially when checking if a prospective zero (like -0.25) is an actual zero.

• It is less time-consuming compared to traditional long division.
• Synthetic division only requires the coefficients of the polynomial, making computation straightforward.

Moreover, finding roots might involve factoring, an algebraic technique that expresses a polynomial as a product of its simpler polynomials. Factoring is particularly helpful as it can easily reveal roots when a polynomial is expressed, for example, as \((x-\text{root})\).
Algebraic manipulation is at the heart of solving polynomials and other equations, and mastering these techniques forms a critical foundation for higher-level mathematics involving calculus and linear algebra.