Step 1: Evaluate .

Step 2. Evaluate

From the previous step, we already know what is.

is just multiplying by another

Step 3: Evaluate .

The expansion of is

Step 1: Evaluate .

Step 2. Evaluate

From the previous step, we already know what is.

is just multiplying by another

Step 3: Evaluate .

The expansion of is

Based upon the corollary to the Fundamental Theorem of Algebra, the degree of a function determines the number of solutions/zeros/roots etc. that exist. They may be real, repeated, imaginary or irrational.

In this case, we must first change the function to be in standard form before determining the degree. Standard form means that the largest exponent goes first and the terms are organized by decreasing exponent.

Now that the polynomial is in standard form, we see that the degree is 8.

There exists 8 total solutions/roots/zeros for this polynomial.

Based upon the corollary to the Fundamental Theorem of Algebra, the degree of a function determines the number of solutions/zeros/roots etc. that exist. They may be real, repeated, imaginary or irrational.

In this case, we must first change the function to be in standard form before determining the degree. Standard form means that the largest exponent goes first and the terms are organized by decreasing exponent.

Now that the polynomial is in standard form, we see that the degree is 8.

There exists 8 total solutions/roots/zeros for this polynomial.

If a complex or imaginary root exists, its' complex conjugate must also exist as a root.

If a complex or imaginary root exists, its' complex conjugate must also exist as a root.

Based upon the fundamental theorem of algebra, we know that there must exist 3 roots for this polynomial based upon its' degree of 3.

To solve for the roots, we use factor by grouping:

First group the terms into two binomials:

Then take out the greatest common factor from each group:

Now we see that the leftover binomial is the greatest common factor itself:

We set each binomial equal to zero and solve:

Step 1: Multiply

Step 2: Multiply the result in step 1 by

Step 3: Multiply the result of step 2 by

Simplify:

Step 1: Multiply

Step 2: Multiply the result in step 1 by

Step 3: Multiply the result of step 2 by

Simplify:

Based upon the fundamental theorem of algebra, we know that there must exist 3 roots for this polynomial based upon its' degree of 3.

To solve for the roots, we use factor by grouping:

First group the terms into two binomials:

Then take out the greatest common factor from each group:

Now we see that the leftover binomial is the greatest common factor itself:

We set each binomial equal to zero and solve: