Lesson 7 Building Strong Roots Solidify Understanding

Jump Start

Which One Doesn’t Belong?

Determine how each of the five equations differs from the others and be prepared to justify your answer with sound mathematical reasoning.

$4 x^{2} - 49 = 0$

$4 x^{2} + 49 = 0$

$x^{2} - 11 x + 24 = 0$

$x^{2} - 11 x + 25 = 0$

$x^{2} - 10 x + 25 = 0$

Learning Focus

Find roots and factors of quadratic and cubic functions.

Write quadratic and cubic equations in factored form.

Identify multiple roots of quadratic and cubic functions.

Do all polynomial functions of degree $n$ have $n$ roots?

Open Up the Math: Launch, Explore, Discuss

When working with quadratic functions, we learned the Fundamental Theorem of Algebra:

An $n^{t h}$ degree polynomial function has $n$ roots.

In this lesson, we will be exploring this idea further with other polynomial functions.

First, let’s brush up on what we learned about quadratics. The equations and graphs of four different quadratic equations are given below. Find the roots for each and identify whether the roots are real or complex/imaginary.

1.

a.

$f (x) = x^{2} + x - 6$

Roots:

Type of roots:

b.

$g (x) = x^{2} - 2 x - 7$

Roots:

Type of roots:

c.

$h (x) = x^{2} - 4 x + 4$

Roots:

Type of roots:

d.

$k (x) = x^{2} - 4 x + 5$

Roots:

Type of roots:

2.

Did all of the quadratic functions have $2$ roots, as predicted by the Fundamental Theorem of Algebra? Explain.

3.

It’s always important to keep what you’ve previously learned in your mathematical bag of tricks so that you can pull it out when you need it. What strategies did you use to find the roots of the quadratic equations?

Pause and Reflect

4.

Using your work from problem 1, write each of the quadratic equations in factored form. When you finish, check your answers by graphing, when possible, and make any corrections necessary.

a.

$f (x) = x^{2} + x - 6$

Factored form:

b.

$g (x) = x^{2} - 2 x - 7$

Factored form:

c.

$h (x) = x^{2} - 4 x + 4$

Factored form:

d.

$k (x) = x^{2} - 4 x + 5$

Factored form:

5.

Based on your work in problem 1, would you say that roots are the same as $x$ -intercepts? Explain.

6.

Based on your work in problem 4, what is the relationship between roots and factors?

Now let’s take a closer look at cubic functions. We’ve worked with transformations of $f (x) = x^{3}$ , but what we’ve seen so far is just the tip of the iceberg. For instance, consider:

$g (x) = x^{3} - 3 x^{2} - 10 x$

7.

Use the graph to find the roots of the cubic function. Use the equation to verify that you are correct. Show how you have verified each root.

8.

Write $g (x)$ in factored form. Verify that the factored form is equivalent to the standard form.

9.

Are the results you found in problem 7 consistent with the Fundamental Theorem of Algebra? Explain.

Here’s another example of a cubic function.

$f (x) = x^{3} + 7 x^{2} + 8 x - 16$

10.

Use the graph to find the roots of the cubic function.

11.

Write $f (x)$ in factored form. Verify that the factored form is equivalent to the standard form. Make any corrections needed.

12.

Are the results you found in problem 10 consistent with the Fundamental Theorem of Algebra? Explain.

13.

We’ve seen the most basic cubic polynomial function, $h (x) = x^{3}$ , and we know its graph looks like this:

Explain how $h (x) = x^{3}$ is consistent with the Fundamental Theorem of Algebra.

14.

Here is one more cubic polynomial function for your consideration. You will notice that it is given to you in factored form. Use the equation and the graph to find the roots of $p (x)$ .

$p (x) = (x + 3) (x^{2} + 4)$

15.

Use the equation to verify each root. Show your work.

16.

Are the results you found in problem 14 consistent with the Fundamental Theorem of Algebra? Explain.

17.

Explain how to find the factored form of a polynomial, given the roots.

18.

Explain how to find the roots of a polynomial, given the factored form.

Ready for More?

Here’s a challenge: Find a cubic function in standard form with real coefficients that has three complex/imaginary roots.

Takeaways

Verify a root:

Roots and $x$ -intercepts:

Finding factored form of a polynomial when the roots are known to be $x = a$ , $x = b$ , $x = c$ :

Multiple roots, or roots of multiplicity $n$ :

Finding roots of a polynomial in factored form:

Vocabulary

Lesson Summary

In this lesson, we found roots of cubic functions using the same methods we learned for quadratic functions. We found that cubic functions can have multiple roots, like quadratic functions. We learned to verify roots and write equivalent equations in factored and standard form. During the lesson, we applied the Fundamental Theorem of Algebra to cubic functions to consider the number and types of possible roots.

Retrieval

1.

Divide: $\begin{array}{r} (x + 7) x^{3} + 5 x^{2} - 5 x + 63 \end{array}$

2.

Use the quadratic formula to find the zeros. $f (x) = 9 x^{2} + 49$