Problem 56

Question

RELATING CONCEPTS For individual or group investigation (Exercises $55-60$ ) The close relationships among $x$ -intercepts of a graph of a function, real zeros of the function, and real solutions of the associated equation should, by now, be apparent to you. Consider the graph of the polynomial function $P(x)=x^{3}-2 x^{2}-11 x+12,$. What are the solutions of the equation $P(x)=0 ?$

Step-by-Step Solution

Verified

Answer

The solutions are $x = 1, 4, -3$.

1Step 1: Understand the Problem

We need to find the real solutions for the equation $P(x)=0$, where $P(x) = x^3 - 2x^2 - 11x + 12$. These solutions correspond to the $x$-intercepts of the graph of the function.

2Step 2: Identify Potential Rational Zeros

Use the Rational Root Theorem to identify possible rational zeros. The possible rational zeros are the factors of the constant term (12) divided by the factors of the leading coefficient (1), which are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$.

3Step 3: Test Potential Zeros

Substitute each potential zero into $P(x)$ to see if it results in zero. For example, try $x = 1$: $P(1) = 1^3 - 2(1)^2 - 11(1) + 12 = 0$. $x = 1$ is a zero of $P(x)$.

4Step 4: Perform Synthetic Division

Use synthetic division to divide $P(x)$ by $x - 1$, since $x = 1$ is a root. This will help break down the polynomial into a simpler form.

5Step 5: Solve the Quadratic Equation

After performing synthetic division, we are left with $x^2 - x - 12$. Solve this quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

6Step 6: Calculate the Solutions

Substituting $a = 1$, $b = -1$, $c = -12$ into the quadratic formula gives: $x = \frac{1 \pm \sqrt{1 + 48}}{2} = \frac{1 \pm \sqrt{49}}{2}$. Thus, $x = 4$ or $x = -3$.

7Step 7: Verify All Solutions

The real solutions of the equation are the x-intercepts: $x = 1, 4, -3$. Verify each solution satisfies $P(x) = 0$. Substitute back into the original equation to confirm.

Key Concepts

Rational Root TheoremSynthetic DivisionQuadratic Formula

Rational Root Theorem

The Rational Root Theorem is an essential tool in mathematics, particularly when dealing with polynomial equations. It provides a systematic way to list all possible rational zeros, making the job of solving polynomial equations significantly easier. According to the theorem, any rational solution of a polynomial equation with integer coefficients can be expressed as $ \frac{p}{q} $, where $ p $ is a factor of the constant term and $ q $ is a factor of the leading coefficient.

For example, if we have a polynomial like $ P(x) = x^3 - 2x^2 - 11x + 12 $, the constant term is 12, and the leading coefficient is 1. Thus, the possible rational roots are the factors of 12, which include:

$ \pm 1 $
$ \pm 2 $
$ \pm 3 $
$ \pm 4 $
$ \pm 6 $
$ \pm 12 $

By testing these values systematically in the polynomial, you can determine which, if any, are actual zeros of the polynomial. This method streamlines the trial and error process, focusing efforts on potentially viable roots.

Synthetic Division

Synthetic division simplifies the process of polynomial division, especially when dividing by linear factors of the form $ x - c $. It is much faster and more efficient than long division for polynomials. Here's how it works:

If you identify that a possible root, like $ x = 1 $, is indeed a zero of your polynomial $ P(x) $, then synthetic division can efficiently divide the polynomial by $ x - 1 $. To perform synthetic division:

Write down the coefficients of the polynomial. For $ P(x) = x^3 - 2x^2 - 11x + 12 $, use 1, -2, -11, and 12.
Place the root you are dividing by, which is 1, to the left of the coefficients.
Perform the synthetic division steps, where each new coefficient is created by multiplying the previous sum by 1 (root), and adding it to the next coefficient.

The result is a reduced polynomial, where $ P(x) $ divided by $ x - 1 $ leaves $ x^2 - x - 12 $. This remainder represents a simpler quadratic equation, which is much easier to solve.

Quadratic Formula

The Quadratic Formula is a universal method for solving quadratic equations. It's invaluable in situations where factorization is difficult or impossible. The formula is expressed as: $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $. This formula can solve any quadratic equation of the form $ ax^2 + bx + c = 0 $.

In our specific problem, after using synthetic division, the polynomial was simplified to $ x^2 - x - 12 $. To use the quadratic formula, identify: