Problem 34 - Algebra 2

Question

One root of the equation \(x^{3}+x^{2}-2=0\) is \(1 .\) What are the other two roots? $$ \begin{array}{lllll}{\text { A. } 1 \pm 2 i} & {\text { B. }-1 \pm i} & {\text { C. } \pm 1+2 i} & {\text { D. } \pm 1-i}\end{array} $$

Step-by-Step Solution

Answer

Using the given root and polynomial division, it is found that the other two roots are -1 and 2.

Step 1: Verify the given root

Substitute the given root, which is x=1, into the equation to verify it as a root. If after substituting, the resulting expression equals zero, the root is correct. For the given equation, this check is: \( (1)^3 + (1)^2 - 2 = 1 + 1 - 2 = 0 \), which confirms that x=1 is indeed a root of the equation.

Step 2: Use Polynomial Division

Since x=1 is a root, we can divide the polynomial by \(x - 1\) (because roots correspond to factors of the form \(x - \text{root value}\)) to find the reduced polynomial of degree 2. Perform polynomial long division or synthetic division to do this.

Step 3: Find the quadratic equation

The polynomial division will give us a quadratic equation that we can solve to find the other two roots. Solving this equation will help to identify the other two roots.

Step 4: Solve the quadratic equation

Use the quadratic formula, factoring, or completing the square to find the solutions to the quadratic equation obtained in step 3. These solutions are the other two roots of the original cubic equation.

Key Concepts

Polynomial Division

Polynomial Division is a mathematical process used to divide a polynomial by another polynomial of lesser or equal degree. Much like long division with numbers, this technique allows us to simplify complex polynomials and find their roots. For example, when a cubic equation has a known root, we can divide the entire polynomial by the binomial associated with this root, which is of the form x - R, where R is the root.

This division reduces the polynomial's degree by one and provides us with a quadratic polynomial. In the context of our exercise, when we discovered that x = 1 is a root, we used polynomial division to divide the cubic equation by x - 1. This yields a simpler equation that can be further analyzed to find the remaining roots.

Roots of Polynomial

Roots of a polynomial are values for which the polynomial equals zero. They are pivotal in understanding the behavior of polynomials, as each root represents an x-intercept on the graph of the polynomial function. A cubic polynomial has exactly three roots, which may be real or complex.

When a root, say x = a, is known, it indicates that (x - a) is a factor of the polynomial. In our exercise, since x = 1 makes the cubic equation zero, it is one of the roots. Finding the other roots involves solving the reduced quadratic equation obtained after removing the known root's factor from the original polynomial.

Synthetic Division

Synthetic Division is an efficient alternative to polynomial long division, especially when dividing by binomials of the form x - R, where R is a constant. This method requires less writing and fewer calculations, making it a favorite for quickly compressing polynomials.

In our example, synthetic division can be used to divide the cubic polynomial by (x - 1) once we verify that x = 1 is a root. The result of synthetic division would be a quadratic polynomial with coefficients that lead us directly to the quadratic equation and hence, closer to finding the remaining roots.

Quadratic Formula

The Quadratic Formula is a powerful tool for solving quadratic equations of the form ax^2 + bx + c = 0. The solutions can be computed using the formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \], where a, b, and c are the coefficients of the quadratic equation. This formula is particularly useful as it provides both real and complex solutions.

In the context of solving our cubic equation, after applying polynomial or synthetic division and obtaining a quadratic equation, we use the Quadratic Formula to find the remaining roots. As indicated by the exercise, the other two roots will be the solutions to the quadratic equation given by the Quadratic Formula.