ACT Math : Polynomials

Example Questions

Example Question #5 : How To Find The Solution To A Binomial Problem

Choose the answer which is a solution to the following binomial equation:

Explanation:

To solve, first pull everything over to one side of the equation:

Then, reduce by dividing by the greatest common denominator—in this case, :

Now, factor:

Therefore:

or

Example Question #6 : How To Find The Solution To A Binomial Problem

Choose the answer below that is a possible solution to the following binomial equation:

Explanation:

The equation presented in the problem is:

To solve this type of equation, you need to factor. First, get all of the terms of the equation on one side:

Then, you need to find two factors that will give you the equation in its current form:

Therefore,  and , so  or .

is listed as an answer, and must therefore be correct.

Example Question #1 : Binomial Denominators

For the equation , what is(are) the solution(s) for ?

Explanation:

, can be factored to (x -7)(x-3) = 0. Therefore, x-7 = 0 and x-3 = 0. Solving for x in both cases, gives 7 and 3.

Example Question #2 : Binomial Denominators

Simplify:

Explanation:

In order to begin this kind of a problem, it's key to look at parts of the rational expression that can be simplified.

In this case, the denominator is an already-simplified binomial; however, the numerator can be factored.

The roots will be numbers that sum up to  but have the product of .

The options include:

When these options are summed up:

We can negate the last three options because the first option of  and  fulfill the requirements. Therefore, the numerator can be factored into the following:

Because the quantity  appears in the denominator, this can be "canceled out." This leaves the final answer to be the quantity .

Example Question #31 : Polynomials

Simplify:

Explanation:

In order to begin this kind of a problem, it's key to look at parts of the rational expression that can be simplified.

In this case, the denominator is an already-simplified binomial; however, the numerator can be factored through "factoring by grouping." This can be a helpful idea to keep in mind when you come across a polynomial with four terms and simplifying is involved.

can be simplified first by removing the common factor of  from the first two terms and the common factor of  from the last two terms:

This leaves two terms that are identical  and their coefficients, which can be combined into another term to complete the factoring:

Consider the denominator; the quantity  appears, so the  in the numerator and in the denominator can be cancelled out. The simplified expression is then left as .

Example Question #1 : How To Simplify Binomials

Solve for x when 6x – 4 = 2x + 5

0

9/4

3

1/4

9/4

Explanation:

Solve by simplifying:

6x – 4 = 2x + 5

6x = 2x + 9

4x = 9

x = 9/4

Example Question #31 : Polynomials

What is the value of the following equation if  and ?

Explanation:

Substitute the numbers 3 and –4 for t and v, respectively.

Example Question #1 : How To Simplify Binomials

Simplify the following binomial:

Explanation:

The equation that is presented is:

To get the correct answer, you first need to combine all of the like terms. So, you can subtract the  from the , leaving you with:

From there, you can reduce the numbers by their greatest common denominator, in this case, :

Then you have arrived at your final answer.

Example Question #1 : Simplifying

Simplify the following binomial:

Explanation:

The equation presented in the problem is:

First you have to combine the like terms, i.e. combining all instances of  and :

Then, you must reduce to find your answer:

Example Question #1 : Simplifying

Simplify the following binomial expression: