Mastery Check Solutions

Show What You Know

The given equation and solutions are incomplete. Determine any additional solutions that are not shown and show all work. Then explain your reasoning.

Incomplete reasoning: $If {(4 x + 1)}^{2} = 81, then x = 2$

$\begin{array}{rcl} \sqrt{{(4 x + 1)}^{2}} & = & \pm \sqrt{81} \\ 4 x + 1 & = & \pm 9 \\ 4 x & = & - 1 \pm 9 \\ x & = & \frac{- 1 \pm 9}{4} \\ x & = & - \frac{10}{4}, \frac{8}{4} \\ x & = & - \frac{5}{2}, 2 \end{array}$

Sample: This equation has two solutions because when you take the square root, you need to remember there can be a positive and negative answer (i.e. ± ). The complete solution has 2 values because of the ± symbol.

Note

Q: What symbol needs to be included when working with the square root property?

A: The plus-minus symbol (±)

The equation is incorrect for the given solutions. Determine a correct equation. Explain where you think the error occurred to get the incorrect equation.

Given $x = i, 3$ , determine a possible polynomial equation with integer coefficients.

Incorrect equation: $x^{2} - 3 x - i x + 3 i = 0$

Sample: The incorrect equation is a second degree polynomial, but it should be a third degree polynomial because imaginary numbers come in conjugate pairs (Conjugate Root Theorem). This means that – i is also a solution. The error is forgetting to include – i as a solution.

$\begin{matrix} x = i, & x = - i, & x = 3 \\ x - i = 0, & x - (- i) = 0, & x - 3 = 0 \end{matrix} (x - i) (x - (- i)) (x - 3) = 0 (x - i) (x + i) (x - 3) = 0 (x^{2} - i^{2}) (x - 3) = 0 (x^{2} - (- 1)) (x - 3) = 0 (x^{2} + 1) (x - 3) = 0 x^{3} - 3 x^{2} + x - 3 = 0$

Note

Remember, if you do not have a third degree polynomial, you need to consider the conjugate pair for imaginary numbers.

Say What You Know

In your own words, talk about what you have learned using the objectives for this lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

Solve polynomial equations by factoring under the set of complex numbers.
Determine the polynomial equation given the roots.
Solve equations using the square root property under the set of complex numbers.