Problem 133 - Introductory Algebra for College Students

Question

Use the \([\mathrm{GRAPH}]\) or \([\mathrm{TABLE}]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If the graphs of \(y_{1}\) and \(y_{2}\) coincide, or if their corresponding table values are equal, this means that the polynomial on the left side has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$x^{3}-1=(x-1)\left(x^{2}-x+1\right)$$

Step-by-Step Solution

Answer

The correct factorization of \(x^{3}-1\) should be \(x^{3}-1=(x-1)(x^{2}+x+1)\). If the graph of \(y_{1}\) (left side: \(x^{3}-1\)) coincides with the graph of \(y_{2}\) (right side: \((x-1)(x^{2}+x+1)\)), the factorization is correct.

Step 1: Graph the Left Side of the Equation

Use a graphing utility to graph the left side of the equation which is given as \(y_{1}=x^{3}-1\). Observe and note down the shape of the graph.

Step 2: Graph the Right Side of the Equation

Next, use your graphing utility to graph the right side of the equation given as \(y_{2}=(x-1)(x^{2}-x+1)\). Observe this graph and compare it with the graph obtained in Step 1.

Step 3: Comparing Graphs \(y_{1}\) and \(y_{2}\)

If the graphs coincide or are superimposed on each other then the polynomial has been correctly factored. If not then it would mean the right side of the equation needs to be factored correctly.

Step 4: If Necessary, Correctly Factor the Polynomial

If the graphs from step 3 do not coincide, this implies that the polynomial \(x^{3}-1\) was not correctly factored. In this case, the correct factorization is \(x^{3}-1=(x-1)(x^{2}+x+1)\).

Step 5: Verification

To confirm that the correct factorization is \(x^{3}-1=(x-1)(x^{2}+x+1)\), plot the new right side as \(y_{2}\) and compare this graph with the graph of \(y_{1}\), which was plotted in step 1. If the two graphs coincide, this confirms the correct factorization.

Key Concepts

Graphing Utility

A graphing utility is an invaluable tool for students dealing with polynomial equations. These electronic aids are like calculators but with extra features that allow you to plot graphs easily. For our problem, we want to use the graphing utility to determine if two polynomials yield the same graph. This involves plotting both sides of the equation.
Here’s how you can do it:

• Enter the first polynomial as function \(y_1 = x^3 - 1\) to see its curve.
• Enter the second polynomial as \(y_2 = (x - 1)(x^2 - x + 1)\) to observe another curve.
• Check if the plots overlap or coincide.

If the graphs of \(y_1\) and \(y_2\) coincide, both sides are identical. If they don't, it indicates an error in factorization, meaning the polynomial needs to be factored correctly. Thus, a graphing utility greatly simplifies visualizing polynomial relationships.

Polynomial Equation

A polynomial equation is an algebraic expression that involves a sum of powers of a variable. In our exercise, the polynomial equation is given as \(x^3 - 1\). The goal of working with such expressions is often to find its roots or to simplify it by factoring it into smaller, more manageable terms.
For the polynomial \(x^3 - 1\):

• The equation is said to be of degree three because the highest power of \(x\) is 3.
• Factoring it requires finding expressions that combine to recreate the original polynomial when multiplied.
• Such factorizations can help find the roots or zeros of the polynomial, which are the values of \(x\) that make the equation zero.

Remember, different sets of factors can represent the same polynomial, but verifying using methods like plotting with a graphing utility ensures accuracy.

Verification of Factors

Verification of factors in polynomial equations is crucial for confirming the correctness of a factorization. This simply means checking if the factorized forms truly multiply back to the original polynomial. To do this, one practical method is to compare graphs generated by a graphing utility.
In our exercise, verification has two important steps:

• If \(y_1 = x^3 - 1\) and \(y_2 = (x-1)(x^2-x+1)\) graphs coincide, initially assume the factorization is correct.
• If they differ, recompute and verify by factoring as \((x-1)(x^2+x+1)\), then reconfirm by plotting and checking overlap again.

Such verification not only boosts understanding but makes sure the factorization can be relied upon for further calculations. Ensuring the polynomials coincide directly checks accuracy and strengthens your grasp on polynomial behavior.