5.3 - Solving Polynomial Equations

5.3 - Solving Polynomial Equations

Objective

To solve polynomial equations by factoring

Recall that we could solve some quadratic equations by factoring the quadratic and using the Zero Product Property

Did you catch this?

This is known as The Factor Theorem

Some Factoring Patterns

Math Response

Factor $$x^3+27$$

Type answer here

Correct Answer: \left(x+3\right)\left(x^2-3x+9\right)

Deg^°

Rad

Here's a thing you may have noticed:

The polynomial was third degree and there were three roots (three factors).

Wow. That is a crazy coincidence. I wonder if it's like that all the time?

Yes. Yes it does.

The number of roots of the polynomial will equal the degree of the polynomial. And the graph of the polynomial will have one fewer turning points than the degree.

Math Response

Factor $$2x^5-18x^4+7x-63$$

Type answer here

Correct Answer: \left(2x^4+7\right)\left(x-9\right)

Deg^°

Rad

Open Ended

Solve the polynomial equation  $$x^3+3x^2-4x-12=0$$  

  

Later on we will learn about the Rational Root Theorem.

So, as a reminder:

If we can factor the polynomial, we can solve the polynomial. We have multiple tactics. We can factor the sum or difference of two cubes if that applies. We can sometimes factor by grouping. If that will not work we'll find out next section that we can use synthetic division to test for divisibility.

Poll

That was a lot. Select the image that most closely matches how you feel about your ability to solve polynomials by factoring: