Algebra II : Non-Quadratic Polynomials

Study concepts, example questions & explanations for Algebra II

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Example Questions

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Example Question #1 : Solutions And Solution Sets

Give all real solutions of the following equation:

Possible Answers:

Correct answer:

Explanation:

By substituting  - and, subsequently,  this can be rewritten as a quadratic equation, and solved as such:

We are looking to factor the quadratic expression as , replacing the two question marks with integers with product  and sum 5; these integers are .

Substitute back:

The first factor cannot be factored further. The second factor, however, can itself be factored as the difference of squares:

Set each factor to zero and solve:

 

Since no real number squared is equal to a negative number, no real solution presents itself here. 

 

The solution set is .

Example Question #1 : Non Quadratic Polynomials

Solve the equation:

Possible Answers:

Correct answer:

Explanation:

 Rewriting the equation as , we can see there are four terms we are working with, so factor by grouping is an apporpriate method. Between the first two terms, the GCF is  and between the third and fourth terms, the GCF is 4. Thus, we obtain.   Setting each factor equal to zero, and solving for , we obtain  from the first factor and  from the second factor. Since the square of any real number cannot be negative, we will disregard the second solution and only accept 

Example Question #2 : Non Quadratic Polynomials

 

What is the degree of the polynomial?

Possible Answers:

Correct answer:

Explanation:

The degree is the highest exponent value of the variables in the polynomial.

Here, the highest exponent is x5, so the degree is 5.

Example Question #1 : Solving Non Quadratic Polynomials

Consider the equation .

According to the Rational Zeroes Theorem, if  are all integers, then, regardless of their values, which of the following cannot be a solution to the equation?

Possible Answers:

Correct answer:

Explanation:

By the Rational Zeroes Theorem, any rational solution must be a factor of the constant, 6, divided by the factor of the leading coefficient, 14. 

Four of the answer choices have this characteristic:

is in lowest terms, and 3 is not a factor of 14. It is therefore the correct answer.

Example Question #3 : Non Quadratic Polynomials

Factor by grouping.

Possible Answers:

Correct answer:

Explanation:

The first step is to determine if all of the terms have a greatest common factor (GCF). Since a GCF does not exist, we can move onto the next step.

Create smaller groups within the expression. This is typically done by grouping the first two terms and the last two terms.

Factor out the GCF from each group:

At this point, you can see that the terms inside the parentheses are identical, which means you are on the right track!

Since there is a GCF of (5x+1), we can rewrite the expression like this:

And that is your answer! You can always check your factoring by FOILing your answer and checking it against the original expression.

Example Question #1 : Non Quadratic Polynomials

Create a cubic function that has roots at .

Possible Answers:

Correct answer:

Explanation:

 

This can be written as:

 

Multiply the terms together:

Multiply the first two terms:

FOIL:

Combine like terms:

Example Question #2 : Non Quadratic Polynomials

Factor completely:

Possible Answers:

The polynomial is prime.

Correct answer:

Explanation:

This can be most easily solved by setting  and, subsequently, . This changes the degree-4 polynomial in  to one that is quadratic in , which can be solved as follows:

The quadratic factors do not fit any factoring pattern and are prime, so this is as far as the polynomial can be factored.

Example Question #6 : Non Quadratic Polynomials

If , and , what is ?

Possible Answers:

Correct answer:

Explanation:

To find , we must start inwards and work our way outwards, i.e. starting with :

We can now use this value to find  as follows:

Our final answer is therefore 

Example Question #2 : Non Quadratic Polynomials

Write a function in standard form with zeroes at -1, 2, and i.

Possible Answers:

Correct answer:

Explanation:

from the zeroes given and the Fundamental Theorem of Algebra we know:

use FOIL method to obtain:

Distribute:

Simplify:

Example Question #8 : Non Quadratic Polynomials

Factor:

Possible Answers:

Correct answer:

Explanation:

Using the difference of cubes formula:

Find x and y:

Plug into the formula:

Which Gives:

And cannot be factored more so the above is your final answer.

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