Algebra II : Polynomials

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #2 : Polynomials

Factor completely: 

Possible Answers:

The polynomial cannot be factored further.

Correct answer:

The polynomial cannot be factored further.

Explanation:

We are looking to factor this quadratic trinomial into two factors, where the question marks are to be replaced by two integers whose product is  and whose sum is 

We need to look at the factor pairs of  in which the negative number has the greater absolute value, and see which one has sum :

None of these pairs have the desired sum, so the polynomial is prime.

Example Question #11 : Variables

Factor completely: 

Possible Answers:

The polynomial cannot be factored further.

Correct answer:

Explanation:

Rewrite this as 

Use the -method by splitting the middle term into two terms, finding two integers whose sum is 1 and whose product is ; these integers are , so rewrite this trinomial as follows:

Now, use grouping to factor this:

 

Example Question #246 : Equations / Inequalities

 

 

Factor the expression:

Possible Answers:

Correct answer:

Explanation:

The given expression is a special binomial, known as the "difference of squares". A difference of squares binomial has the given factorization: . Thus, we can rewrite  as  and it follows that 

Example Question #247 : Equations / Inequalities

Factor the equation:

Possible Answers:

Correct answer:

Explanation:

The product of is .

For the equation

must equal and  must equal .

Thus  and must be and , making the answer  .

Example Question #1 : How To Factor The Quadratic Equation

Find solutions to .

Possible Answers:

Correct answer:

Explanation:

The quadratic can be solved as . Setting each factor to zero yields the answers.

Example Question #41 : How To Factor A Polynomial

Factor:

Possible Answers:

The expression cannot be factored.

Correct answer:

Explanation:

Because both terms are perfect squares, this is a difference of squares:

The difference of squares formula is .

Here, a = x and b = 5.  Therefore the answer is .

You can double check the answer using the FOIL method:

Example Question #41 : How To Factor A Polynomial

Factor:

Possible Answers:

Correct answer:

Explanation:

The solutions indicate that the answer is:

and we need to insert the correct addition or subtraction signs. Because the last term in the problem is positive (+4), both signs have to be plus signs or both signs have to be minus signs. Because the second term (-5x) is negative, we can conclude that both have to be minus signs leaving us with:

Example Question #41 : Factoring Polynomials

Factor the following polynomial: .

Possible Answers:

Correct answer:

Explanation:

Because the term doesn’t have a coefficient, you want to begin by looking at the  term () of the polynomial: .  Find the factors of  that when added together equal the second coefficient (the term) of the polynomial. 

There are only four factors of : , and only two of those factors, , can be manipulated to equal  when added together and manipulated to equal  when multiplied together: (i.e.,). 

Example Question #42 : Factoring Polynomials

Factor the following polynomial: .

Possible Answers:

Correct answer:

Explanation:

Because the  term doesn’t have a coefficient, you want to begin by looking at the  term () of the polynomial: 

Find the factors of  that when added together equal the second coefficient (the  term) of the polynomial:

There are seven factors of , and only two of those factors, , can be manipulated to equal  when added together and manipulated to equal  when multiplied together:  

Example Question #5 : How To Find The Degree Of A Polynomial

Simplify:

Possible Answers:

1

5

None of the above

2x

-1

Correct answer:

-1

Explanation:

The given expression can be re-written as:

Cancel (2x - 5):

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