Problem 120 - Introductory Algebra for College Students

Question

Use a graphing utility to graph each side of the equation in the same viewing rectangle. Do the graphs coincide? If so, this means that the polynomial on the left side has been factored correctly. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$x^{2}+2 x+x+2=x(x+2)+1$$

Step-by-Step Solution

Answer

The polynomial \(x^{2}+3x+2\) is correctly factored as \((x+1)(x+2)\). This can be verified by using a graphing utility that will show coinciding graphs for \((x+1)(x+2)\) and \(x(x+2)+1\).

Step 1: Graphing the Original Equation

The equation is \(x^{2}+2 x+x+2\). This is graphed to see its shape and relation to the other part of the equation. The same process is done for \(x(x+2)+1\).

Step 2: Comparing Graphs

Compare both graphs obtained to check if they coincide. If they do, it is inferred that the polynomial factorization was done correctly. However, if the graphs do not coincide, proceed to the next step.

Step 3: Factoring the Polynomial

To correctly factor the polynomial, we group similar terms, and then factor the common elements. Here \(x^{2}+2 x+x+2\) = \(x^{2}+3x+2\). To factor correctly, we find two numbers that multiply to the constant term (2) and add to the coefficient of the x term (3), which are 1 and 2. Therefore, the polynomial can be factored as \((x+1)(x+2)\).

Step 4: Verification Using the Graphing Utility

After factoring the polynomial correctly, we use the graphing utility to graph the factorized function, \((x+1)(x+2)\), and the other part of the equation \(x(x+2)+1\). If the graphs coincide, the polynomial has been factored correctly.