Problem 27 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&x^{3}+x^{2}-7 x-7=0\\\&[-10,10] \text { by }[-20,20]\end{aligned}$$

Step-by-Step Solution

Answer

The complex solutions are \(x = -1, \sqrt{7}, -\sqrt{7}\).

Step 1: Identify the type of polynomial

The given equation is a cubic polynomial: \(x^3 + x^2 - 7x - 7 = 0\). This indicates there can be at most three complex solutions, including any real solutions.

Step 2: Attempt to factor the polynomial

First, try to factor the polynomial by checking for any rational roots using the Rational Root Theorem. The potential rational roots are the factors of the constant term, \(-7\), which are \(\pm 1, \pm 7\).

Step 3: Evaluate potential rational roots

Evaluate the polynomial for these potential roots: - \(f(1) = 1^3 + 1^2 -7 \times 1 - 7 = -12\) - \(f(-1) = (-1)^3 + (-1)^2 - 7(-1) - 7 = 0\)- \(f(7) = 7^3 + 7^2 - 7\times7 - 7 = 294\)- \(f(-7) = (-7)^3 + (-7)^2 - 7(-7) - 7 = -294\) Thus, \(x = -1\) is a root.

Step 4: Perform polynomial division

Since \(x = -1\) is a root, factor \((x+1)\) out of the polynomial using synthetic or long division. Dividing \(x^3 + x^2 - 7x - 7\) by \(x + 1\) gives a quotient of \(x^2 - 7\) and a remainder of 0.

Step 5: Solve the quadratic equation

The remaining factor is \(x^2 - 7 = 0\). Solving this quadratic equation, we add 7 to both sides: \(x^2 = 7\). Taking the square root, we get \(x = \pm\sqrt{7}\).

Step 6: Compile the solution set

The complete solution set consists of the roots found: - \(x = -1\), - \(x = \sqrt{7}\), - \(x = -\sqrt{7}\). Thus, the solution set is \(\{-1, \sqrt{7}, -\sqrt{7}\}\).

Step 7: Graph the function and verify real solutions

Graph the function \(Y_1 = x^3 + x^2 - 7x - 7\) in the specified window \([-10, 10] \) by \([-20, 20]\). Use a calculator to confirm the real solutions. The graph should cross the x-axis at the real solutions \(x = -1\), \(x = \sqrt{7}\), and \(x = -\sqrt{7}\), supporting our solution set.

Key Concepts

Cubic Polynomial

A cubic polynomial is a type of polynomial equation with the highest degree of three. In other words, it is an equation of the form \( ax^3 + bx^2 + cx + d = 0 \), where \( a \), \( b \), \( c \), and \( d \) are constants, and \( a eq 0 \). This ensures that the term with \( x^3 \) is present, making it a cubic equation. The highest exponent on the variable \( x \) dictates the number of solutions, or roots, the polynomial can have. Thus, a cubic polynomial can have up to three solutions. These solutions might include a mix of real and complex roots.

Understanding cubic polynomials is crucial, as their roots often represent important, meaningful solutions such as points of intersection or equilibrium in real-world scenarios. Unlike quadratic equations, there are no general formulas for the roots of cubic polynomials, making factorization, and methods like the Rational Root Theorem, essential tools.

Rational Root Theorem

The Rational Root Theorem is a powerful tool when solving polynomial equations, including cubic polynomials. It helps identify possible rational solutions, that is, solutions that can be expressed as fractions. According to this theorem, any possible rational root of a polynomial equation \( ax^n + bx^{n-1} + \, ... \, + k = 0 \) will be of the form \( \frac{p}{q} \). Here, \( p \) is a factor of the constant term \( k \), and \( q \) is a factor of the leading coefficient \( a \).

Applying this theorem efficiently narrows down the list of potential rational roots. In the case of the equation \( x^3 + x^2 - 7x - 7 = 0 \), the constant term is -7, whose factors include \( \pm1 \) and \( \pm7 \). Testing these values helped identify \( x = -1 \) as a root. This theorem is especially useful before attempting more complex methods, like polynomial division, as it can quickly conclude the presence of simple rational roots.

Polynomial Division

Polynomial division is a technique used to divide one polynomial by another, simplifying equations by factoring out known roots. It is similar to the long division process used in arithmetic. There are two main forms: synthetic division and long division. In this context, they help break down higher degree polynomials after finding a root.

When \( x = -1 \) was confirmed as a root of \( x^3 + x^2 - 7x - 7 = 0 \), polynomial division allowed us to divide the original polynomial by \( x + 1 \). The process revealed a quotient of \( x^2 - 7 \) with no remainder, confirming the factorization. This step simplifies the original cubic polynomial into smaller, more manageable quadratic terms, paving the way for further analysis using simpler techniques like solving quadratic equations.

Quadratic Equation

Once a polynomial is reduced to a quadratic form through methods such as factoring or polynomial division, solving becomes relatively straightforward. A quadratic equation is any equation that can be written in the format \( ax^2 + bx + c = 0 \). Solutions to quadratic equations can be found via methods such as factoring, completing the square, or using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

In our problem, after dividing \( x^3 + x^2 - 7x - 7 \) by \( x+1 \), we obtained \( x^2 - 7 = 0 \). This can be easily solved by isolating \( x^2 \) and taking the square root of both sides, resulting in \( x = \pm\sqrt{7} \). Quadratic equations are common across various mathematical applications, and mastering them simplifies solving more complex problems like cubic polynomials.