Problem 56 - Algebra for College Students

Question

With a graphing calculator an equation can be solved without the kind of hint that was given for Exercises 43–52. Solve each of the following equations by examining the graph of a corresponding function. Use synthetic division to check. a) \(x^{3}-4 x^{2}-7 x+10=0\) b) \(8 x^{3}-20 x^{2}-18 x+45=0\)

Step-by-Step Solution

Answer

Graph the functions and identify the roots where the graphs intersect the x-axis. Confirm roots using synthetic division and solve any remaining polynomial.

Step 1 - Graph the function

For each equation, convert it into a function and graph it using a graphing calculator.For (a), graph the function: \[ f(x) = x^{3} - 4 x^{2} - 7 x + 10 \]For (b), graph the function: \[ g(x) = 8 x^{3} - 20 x^{2} - 18 x + 45 \]Observe where the graph crosses the x-axis to find the roots of the equations.

Step 2 - Identify the roots from the graph

By examining the graphs:(a) For \( f(x) = x^{3} - 4 x^{2} - 7 x + 10 \), identify the x-values where the graph touches or crosses the x-axis. These x-values are the roots of the equation.(b) For \( g(x) = 8 x^{3} - 20 x^{2} - 18 x + 45 \), same as above, identify the x-values where the graph crosses the x-axis.

Step 3 - Confirm roots using synthetic division

Use synthetic division to verify the identified roots: (a) Let's assume one of the roots found from the graph for \( x^{3} - 4x^{2} - 7x + 10 = 0 \) is \( x = a \). Perform synthetic division to see if the remainder is 0.(b) Perform synthetic division on \( 8x^{3} - 20x^{2} - 18x + 45 = 0 \) with the assumed root \( x = b \). If the remainder is zero, \( x = b \) is a root.

Step 4 - Solve remaining polynomial (if needed)

If synthetic division confirms a root, continue solving the resulting polynomial until all roots are identified.For example, if \( x = a \) is confirmed for (a), we factor \( (x - a) \) from the polynomial and solve the resulting quadratic.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool to visually solve polynomial equations like cubic equations. To use a graphing calculator for solving these equations, follow these steps:

1. Inputting the Function:
First, rewrite the given equation as a function. For example, if you have the equation \(x^{3} - 4x^{2} - 7x + 10 = 0\), input it as the function \(f(x) = x^{3} - 4x^{2} - 7x + 10\).

2. Graphing the Function:
Next, use the graphing feature to plot the function. The graph will show a curve representing the polynomial. The points where the graph intersects the x-axis are the roots of the equation. Each intersection denotes an x-value for which \(f(x) = 0\).

3. Identifying Roots:
Closely examine the x-intercepts. Note down these x-values, as they represent the approximate roots of your polynomial equation. For instance, if the graph of \(f(x)\) crosses the x-axis at x = 1, x = -2, and x = 3, these are your potential solutions.

Synthetic Division

Synthetic division is a simplified form of dividing polynomials, particularly useful for verifying potential roots obtained from graphing. Here's how you can use synthetic division:

1. Set Up the Division:
Write down the coefficients of the polynomial and an assumed root. For example, with the polynomial \(x^{3} - 4x^{2} - 7x + 10\) and a root candidate x = 2, list the coefficients 1, -4, -7, 10.

2. Perform Synthetic Division:
Start by bringing down the first coefficient (1). Multiply this by the root (2), and write the result under the second coefficient (-4). Add these numbers to get a new coefficient. Repeat this process for all terms in the polynomial.

3. Verify the Root:
If the final number (remainder) is zero, the assumed x-value is a root. If it isn’t zero, the value is not a root. For the polynomial \(x^{3} - 4x^{2} - 7x + 10\) with an assumed root x = 2, if the remainder after synthetic division is zero, x = 2 is confirmed as a root.

Roots of Polynomial Equations

Finding the roots of polynomial equations is fundamental in algebra. Here’s how you can methodically solve for the roots:

1. Graphing Polynomials:
Use a graphing calculator to visualize the polynomial graph. The x-values where the graph intersects the x-axis are your initial root estimates. These intersections point to the real roots of your equation.

2. Confirming Roots:
Once you have potential roots from graphing, employ synthetic division to verify these roots. If the root is correct, synthetic division will result in a remainder of zero.

3. Solving Remaining Polynomials:
If a root is confirmed, factor it out from the polynomial and simplify. This may lead to a quadratic or smaller degree polynomial, which can be solved using standard techniques like factoring, completing the square, or the quadratic formula.

For example, if solving the cubic polynomial \(x^{3} - 4x^{2} - 7x + 10\) reveals x = 2 as a root, you then factorize it to get \((x - 2)(x^{2} - 2x - 5)\). Solve the quadratic equation \(x^{2} - 2x - 5\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\).

By carefully following these steps, you can find all roots of any polynomial equation efficiently.