Polynomials - GRE Subject Test: Math

Solution:

We can look at Pascal's Triangle, which is a quick way to do Binomial Expansion. We read each row (across, left to right)

For the first row, we only have a constant.
For the second row, we get .
...
For the 7th row, we will start with an term and end with a constant.

Step 1: We need to locate the 7th row of the triangle and write the numbers in that row out.

The 7th row is .

Step 2: If we translate the 7th row into an equation, we get:

. This is the solution.

Method One:

We will start expanding slowly, and we will end up at exponent

Step 1: Expand:

Step 2: Multiply by the product of . By doing this, we are now expanding .

Step 3: Multiply by again

.

After Step 4:
After Step 5:

After Step 6, the final answer is:

.

Method Two:

You can find the expansion of this binomial by using the Pascal's Triangle (shown below)

If you look at Row of the triangle above, the row that starts with .

We need to negate every nd term, as the answer in Method One has every even term negative.

We will still get the answer: .

Step 1: Expand

Step 2: FOIL the first two parentheses:

Step 3: Multiply the expansion in step 2 by :

The expanded form of is .

Step 1: Multiply

Step 2: Multiply the result in step 1 by

Step 3: Multiply the result of step 2 by

Simplify:

Expand by distributing each of the factors

Simplify

Simplify

Step 1: Let's start small, expand .

Step 2: Expand

Take the final answer of Step 1 and multiply it by ...

Step 3: Multiply again by to the final answer of Step 2...

Let's start with a smaller expansion:

We multiply the expansion of by :

Multiply again by :

Multiply by :

The easiest way to expand binomials raised to higher powers is to use Pascal's Triangle.

Pascal's Triangle is used to find the multipliers for each level of exponent.

It follows the pattern listed below.

To complete the expansion we will take the row that corresponds to the 4th exponent for this problem.

We will now organize this into columns and rows.

The second and third rows are organized by taking the left term from the highest to lowest power, and the right term from lowest to highest power.

Since the x is on the left, it is raised to the 4th power, 3rd power and so on.

Since the 3 is on the right, it is raised to the 0 power, 1 power and so on.

We now simplify each of the terms, the bottom row is the only one to be simplified in this case. Anything to the zero power except zero is 1.

Now we multiply each column together to obtain the full expansion.

For example to obtain the third term we multiplied everything in the 3rd column:

We did this for all of the columns to get the below final answer.

The corollary to the Fundamental Theorem of Algebra states that for any polynomial the number of solutions will match the degree of the function.

The degree of a function is determined by the highest exponent for x, which in this case is 7.

This means that there will be 7 solutions total for the below function.

This means that max number of REAL solutions would be 7, but the total number of solutions, real, repeated or irrational will total 7.

In order to determine the correct answer, we must first change the function to be in standard form. A function in standard form begins with the largest exponent then decreases from there.

We must change:

to become:

Once we have established standard form, we can now see that this is a degree 3 polynomial, which means that it will have 3 roots or solutions.

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

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.

Step 1: Find factors of 44:

Step 2: Find which pair of factors can give me the middle number. We will choose .

Step 3: Using and , we need to get . The only way to get is if I have and .

Step 4: Write the factored form of that trinomial:

Step 5: To solve for x, you set each parentheses to :

The solutions to this equation are and .

Step 1: Factor by pairs:

Step 2: Re-write the factorization:

Step 3: Solve for x:

Step 1: Find two numbers that multiply to and add to .

We will choose .

Step 2: Factor using the numbers we chose:

Step 3: Solve each parentheses for each value of x..

Step 1: Find two numbers that multiply to and add to ...

We will choose

Check:

We have the correct numbers...

Step 2: Factor the polynomial...

Step 3: Set the parentheses equal to zero to get the roots...

So, the roots are .

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: