High School Math : Polynomials

Study concepts, example questions & explanations for High School Math

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Example Questions

Example Question #1 : Understanding Zeros Of A Polynomial

Factor the polynomial if the expression is equal to zero when .

Possible Answers:

Correct answer:

Explanation:

Knowing the zeroes makes it relatively easy to factor the polynomial.

The expression fits the description of the zeroes.

Now we need to check the answer.

We are able to get back to the original expression, meaning that the answer is .

Example Question #1 : Understanding Zeros Of A Polynomial

A polyomial with leading term  has 6 as a triple root. What is this polynomial?

Possible Answers:

 

Correct answer:

Explanation:

Since 6 is a triple root, and the degree of the polynomial is 3, the polynomial is , which we can expland using the cube of a binomial pattern.

Example Question #1 : Express A Polynomial As A Product Of Linear Factors

A polyomial with leading term  has 5 and 7 as roots; 7 is a double root. What is this polynomial?

Possible Answers:

Correct answer:

Explanation:

Since 5 is a single root and 7 is a double root, and the degree of the polynomial is 3, the polynomial is . To put this in expanded form:

 

Example Question #3 : Understanding Zeros Of A Polynomial

What are the solutions to  ? 

Possible Answers:

Correct answer:

Explanation:

When we are looking for the solutions of a quadratic, or the zeroes, we are looking for the values of  such that the output will be zero. Thus, we first factor the equation. 

Then, we are looking for the values where each of these factors are equal to zero. 

 implies 

and  implies 

Thus, these are our solutions. 

Example Question #1 : Finding Zeros Of A Polynomial

Find the zeros of the following polynomial:

Possible Answers:

Correct answer:

Explanation:

First, we need to find all the possible rational roots of the polynomial using the Rational Roots Theorem:

Since the leading coefficient is just 1, we have the following possible (rational) roots to try:

±1, ±2, ±3, ±4, ±6, ±12, ±24

When we substitute one of these numbers for , we're hoping that the equation ends up equaling zero. Let's see if  is a zero:

Since the function equals zero when  is , one of the factors of the polynomial is . This doesn't help us find the other factors, however. We can use synthetic substitution as a shorter way than long division to factor the equation.

            

      

Now we can factor the function this way:

We repeat this process, using the Rational Roots Theorem with the second term to find a possible zero. Let's try :

When we factor using synthetic substitution for , we get the following result:

Using our quadratic factoring rules, we can factor completely:

Thus, the zeroes of  are

Example Question #1 : Simplifying Polynomial Functions

Simplify the following polynomial:

Possible Answers:

Correct answer:

Explanation:

To simplify the polynomial, begin by combining like terms:

Example Question #1 : Polynomials

Simplify the following polynomial function:

Possible Answers:

Correct answer:

Explanation:

First, multiply the outside term with each term within the parentheses:

Rearranging the polynomial into fractional form, we get:

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