Problem 27 - A Graphical Approach to College Algebra

Question

Solve each equation analytically 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 your calculator, support 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

Solutions: \(x = -1\), \(x = \sqrt{7}\), and \(x = -\sqrt{7}\). Verify real solutions using a graph.

Step 1: Identify the polynomial equation

The given equation is a cubic polynomial: \(x^3 + x^2 - 7x - 7 = 0\). We need to find all complex solutions for \(x\).

Step 2: Use Rational Root Theorem

Apply the Rational Root Theorem to find potential rational roots. The possible rational roots are factors of the constant term \(-7\), which are \(\pm 1, \pm 7\).

Step 3: Test potential rational roots

Check each possible root by substituting them into the equation. We test \(x = 1\): \(1^3 + 1^2 - 7(1) - 7 = -12\). Test \(x = -1\): \((-1)^3 + (-1)^2 - 7(-1) - 7 = 0\), so \(x = -1\) is a root.

Step 4: Factor the polynomial

Use \(x = -1\) to factor the polynomial. Perform synthetic division with \(x + 1\) and the original polynomial \(x^3 + x^2 - 7x - 7\), resulting in \(x^2 - 7\). Thus, the polynomial factored form is \((x + 1)(x^2 - 7) = 0\).

Step 5: Solve the factored equation

Solve \((x + 1)(x^2 - 7) = 0\). From \(x + 1 = 0\), we get \(x = -1\). From \(x^2 - 7 = 0\), solve for \(x^2 = 7\), giving \(x = \sqrt{7}\) and \(x = -\sqrt{7}\).

Step 6: List all complex solutions

The complete set of solutions for the equation \(x^3 + x^2 - 7x - 7 = 0\) is \(x = -1\), \(x = \sqrt{7}\), and \(x = -\sqrt{7}\).

Step 7: Graph the polynomial

Graph the function \(y_1 = x^3 + x^2 - 7x - 7\) using a calculator within the window \([-10, 10]\) for \(x\) and \([-20, 20]\) for \(y\).

Step 8: Verify real solutions with graph

Check the graph intersections with the x-axis. Real solutions occur at \(x = -1\), \(x = -\sqrt{7}\), and \(x = \sqrt{7}\), which should match the x-intercepts seen on the graph.

Key Concepts

Cubic Polynomial

A cubic polynomial is a type of polynomial with the highest degree being three. In simpler terms, it means the equation includes an expression with a variable raised to the power of three. An example of a cubic polynomial is: \[x^3 + x^2 - 7x - 7\]. Cubic polynomials are more complex than linear and quadratic polynomials because they have up to three solutions, which can be real or complex numbers. They can be expressed in the general form:

• \(ax^3 + bx^2 + cx + d = 0\)

where "a," "b," "c," and "d" are constants, and "a" is not equal to zero. Finding the solutions requires various techniques, such as factoring and using the Rational Root Theorem.

Rational Root Theorem

The Rational Root Theorem is a useful tool in algebra for finding rational solutions to polynomial equations. It helps identify potential roots based on the relationship between the coefficients and the constant term of the polynomial. For a cubic equation like \(x^3 + x^2 - 7x - 7 = 0\), the theorem advises us to check divisors of the constant term,

• In this case, the constant term is \(-7\).
• The possible rational roots might be \(\pm 1, \pm 7\).

By substituting these possible roots into the polynomial, we can verify whether they satisfy the equation. For example, substituting \(x = -1\) into the equation confirms that it is a root, as it balances the equation to zero. This step narrows down the options and helps proceed with more advanced methods like synthetic division.

Synthetic Division

Synthetic division is a simplified method of dividing polynomials, typically used for dividing by linear factors. It's less cumbersome than long division and helps find factors of polynomials more efficiently. Once we identify one rational root using the Rational Root Theorem, like \(x = -1\), synthetic division allows us to break down the polynomial further. In our cubic polynomial \(x^3 + x^2 - 7x - 7\), known root \(x = -1\) helps us factor the polynomial. Performing synthetic division with \(x + 1\) yields:

• A quotient of \(x^2 - 7\).

This process transforms the original cubic polynomial into a simpler form, making the remaining roots easier to identify through simpler methods such as solving quadratic equations.

Graphical Analysis

Graphical analysis is an effective way to visualize solutions to polynomial equations. By graphing the left side of the original equation as a function, we can identify where the graph intersects the x-axis, revealing real roots of the equation. For the polynomial \(y = x^3 + x^2 - 7x - 7\), graphing it within the window

• \([-10, 10]\) for \(x\) and \([-20, 20]\) for \(y\)

provides crucial visual insights. The x-intercepts on this graph correspond to the real solutions of the equation. By confirming these visual results against previously calculated solutions \((x = -1, x = \sqrt{7}, x = -\sqrt{7})\), we can confidently state that they are indeed accurate. This visual verification complements the analytical methods, offering a comprehensive approach to solving the polynomial.