# Lesson 4Getting to the Root of the ProblemSolidify Understanding

## Jump Start

Graph each function:

## Learning Focus

Break down a polynomial function to find the roots.

Write a polynomial function in factored form.

How can we find all the factors and roots of a polynomial?

## Open Up the Math: Launch, Explore, Discuss

Previously, we learned to predict the number of roots of a polynomial using the Fundamental Theorem of Algebra and the relationship between roots and factors. In this lesson, we will be working on how to find all the roots of a polynomial given in standard form.

Let’s start by reviewing numbers and factors.

### 1.

If you know that is a factor of , what would you do to find the prime factorization of ? Explain your answer and show your process here:

### 2.

How is your answer like a polynomial written in the form: ?

The process for finding factors of polynomials is exactly like the process for finding factors of numbers. We start by dividing by a factor we know and keep dividing until we have all of the factors. When we break down the polynomial into a quadratic, sometimes we can factor it by inspection, and sometimes we can use other quadratic tools such as the quadratic formula.

Let’s try it! For each of the following functions, you have been given one factor. Use that factor to find the remaining factors, determine the roots of the function, and write the function in factored form.

### 3.

Function:

Factor:

Roots of function:

Factored form:

Pause and Reflect

### 4.

Function:

Factor:

Roots of function:

Factored form:

### 5.

Function:

Factor:

Roots of function:

Factored form:

### 6.

Function:

Factor:

Roots of function:

Factored form:

### 7.

Function:

Factor:

Roots of function:

Factored form:

### 8.

Function:

Factor:

Roots of function:

Factored form:

### 9.

Is it possible for a polynomial with real coefficients to have only one non-real complex root? Explain.

### 10.

Based on the Fundamental Theorem of Algebra and the polynomials that you have seen, make a table that shows all of the number of roots and the possible combinations of real and non-real complex roots for linear, quadratic, cubic, and quartic polynomials.

Do the irrational roots of polynomials always occur in conjugate pairs? Explain your answer.

## Takeaways

Type of Polynomials

# of Roots

Types of Roots

Linear – Degree

Cubic – Degree

Quartic – Degree

## Lesson Summary

In this lesson, we learned to find the roots of a polynomial by using long division and factoring to break down the polynomial into factors of lower degree. Technology was useful for checking work and finding real roots that can be used to find complex roots. We determined the number and type of roots possible for each polynomial, which will be useful in writing equations and finding roots in upcoming lessons.

## Retrieval

### 1.

Order the numbers from least to greatest.