# GRE Subject Test: Math : Roots of Polynomials

## Example Questions

### Example Question #1 : Roots Of Polynomials

What are the roots of the polynomial: ?

None of the Above

Explanation:

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 .

### Example Question #1 : Algebra

Solve for

Explanation:

Step 1: Factor by pairs:

Step 2: Re-write the factorization:

Step 3: Solve for x:

### Example Question #1 : Algebra

Find

No Solutions Exist

Explanation:

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..

### Example Question #1 : Algebra

Explanation:

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:

### Example Question #1 : Polynomials

Find all of the roots for the polynomial below:

Explanation:

In order to find the roots for the polynomial we must first put it in Standard Form by decreasing exponent:

Now we can use factor by grouping, we start by grouping the 4 terms into 2 binomials:

We now take the greatest common factor out of each binomial:

We can see that each term now has the same binomial as a common factor, so we simplify to get:

To find all of the roots, we set each factor equal to zero and solve:

Explanation:

### Example Question #1 : Polynomials

What are the roots of ?

Explanation:

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 .

### Example Question #1 : Classifying Algebraic Functions

Find all of the roots for the polynomial below:

Explanation:

In order to find the roots of the polynomial we must factor by grouping:

Group into two binomials:

Take out the greatest common factor from each binomial:

We can now see that each term has a common binomial factor:

We set each factor equal to zero and solve to obtain the roots: