Problem 38 - College Algebra

Question

Use synthetic division to divide \(f(x)=x^{3}-2 x^{2}\) \(x+2\) by \(x+1 .\) Use the result to find all zeros of \(f\).

Step-by-Step Solution

Answer

The division function by using synthetic method is: \(x^{2} -3x + 3 - \frac{1}{x+1}\). And the zeros of the function \(f(x) = x^{3} - 2x^{2} + 0x +2\) are \(1\) and \(x=\frac{1 \pm \sqrt{5}}{2}\)

Step 1: Set up for Synthetic Division

First, setup the synthetic division with the coefficients of the polynomial function at the top and the divisor at the bottom. In this case, the polynomial function is \(f(x)=x^{3}-2x^{2}+0x+2\) (note: there is a 0 for \(x\) because there is no \(x\) term in the original function) and the divisor is \(x+1\) where divisor is -1 because we have \(+1\) in the divisor.

Step 2: Perform Synthetic Division

Carry down the leading coefficient to the bottom row. Multiply this number by the divisor and write the result under the next coefficient. Add the numbers in this column and write the result below these numbers. Repeat until all columns are complete. Here the coefficients for the bottom row are: [1, -3, 3, -1]. The last coefficient is the remainder of the division.

Step 3: Write the Result of the Division as a Polynomial

The result of the synthetic division can be written as another polynomial. The degree of this polynomial is always one less than the degree of the divisor. The coefficients of the new polynomial are the numbers in the bottom row of the synthetic division. In this case the result is \(x^{2} - 3x + 3\). The remainder is \(-1\). Thus the division result will be: \(x^{2} -3x + 3 - \frac{1}{x+1}\).

Step 4: Finding the Zeros of the Function \(f\)

Zeros of the function are found by setting \(f(x) = 0\) and solving for \(x\). Substitute the function \(f(x)\) with the result from the division which is easier to solve. So the zeros are the solutions for the equation: \(f(x) = x^{2} -3x + 3 - \frac{1}{x+1} = 0\). After simplifying, we're left with equation \(x^{3} - 4x^{2} + 4x -1 = 0\). Factoring we got that the zeros are \(1\) and \(x=\frac{1 \pm \sqrt{5}}{2}\)

Key Concepts

Polynomial Division

Polynomial division is a method to divide one polynomial by another, similar to how we divide numbers in long division. It allows us to find a quotient and sometimes a remainder, much like when dividing integers. In the case of our exercise, we used synthetic division, a shorthand version of polynomial division particularly useful when dividing by linear divisors like \(x + 1\).

It involves using the coefficients of the polynomial, simplifying the entire process to a series of multiplication and addition operations. When applying synthetic division, you place the coefficients of the polynomial in a row and perform the steps of the method. By proceeding with arithmetic operations systematically, you arrive at a new polynomial that is one degree less than the original. This new polynomial and any remainder express the division in a simplified manner, as shown in the exercise solution where we obtained \(x^2 - 3x + 3 - \frac{1}{x+1}\). By doing so, it sets the stage for discovering the roots of the polynomial.

Roots of Polynomials

The roots, or zeros, of a polynomial are the solutions to the equation \(f(x) = 0\). Finding these roots is essential because they indicate where the polynomial graph intersects the x-axis. They can be real or complex, and often provide a wealth of information about the polynomial itself.

In our exercise, after performing synthetic division, we reduced the original polynomial to a simpler quadratic equation \(x^2 - 3x + 3\). By factoring or using the quadratic formula on this expression, we found the roots. These roots not only help solve the equation but also offer insights about the graph and behavior of the polynomial function.

It's important to remember that a polynomial of degree \(n\) will have \(n\) roots, though some may be repeated or non-real. Understanding the nature of these roots is crucial in applications across various fields of science and engineering.

Zero Finding

Zero finding, often synonymous with root finding, involves solving \(f(x) = 0\) for its zeros. The zeros represent the x-values where the polynomial equals zero. For polynomials, zeros correspond to the roots, which we explored via synthetic division.

Our example utilized synthetic division to simplify the process and obtain a simplified polynomial that was easier to solve. By setting this new polynomial equal to zero, we efficiently located all zeros of the original function \(f(x)\).

Methods for zero finding include factoring, using the quadratic formula, and graphing approaches, each providing different insights into the polynomial's nature. These methods help in identifying whether zeros are real or complex, repeated or distinct, each having implications for the polynomial's graph and applications.

Quadratic Formula

The quadratic formula is a reliable tool to find the roots of quadratic equations of the form \(ax^2 + bx + c = 0\). Given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), it calculates the roots by working directly from the quadratic coefficients.

In our exercise, after reducing the cubic polynomial to a quadratic, the quadratic formula was used to find the roots of \(x^2 - 3x + 3 = 0\).

The quadratic formula provides an efficient means to solve for roots when factoring isn't easily apparent. It leverages the discriminant, \(b^2 - 4ac\), to indicate the nature of the roots – whether they are real or complex. Its use is particularly vital in ensuring all roots are found accurately, making it indispensable in both theoretical and applied mathematical contexts.