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Algebra II : Polynomial Functions

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #1 : How To Find The Degree Of A Polynomial

Give the degree of the polynomial.

$6x^{5}+5x^{4}-3x^{2}+x^{7}$

Possible Answers:

Correct answer:

Explanation:

$6x^{5}+5x^{4}-3x^{2}+x^{7}$

The degree of an individual term of a polynomial is the exponent of its variable; the exponents of the terms of this polynomial are, in order, 5, 4, 2, and 7.

The degree of the polynomial is the highest degree of any of the terms; in this case, it is 7.

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Example Question #1 : How To Find The Degree Of A Polynomial

What is the degree of the polynomial?

$ab^{2}+a^{2}b^{3}-a^{3}b^{1}$

Possible Answers:

Correct answer:

Explanation:

$ab^{2}+a^{2}b^{3}-a^{3}b^{1}$

To find the degree of the polynomial, you first have to identify each term [term is for example $ab^{2}$ ], so to find the degree of each term you add the exponents.

EX: $ab^{2}$ $= a^{1}b^{2} = 1+2= 3$ - Degree of 3

Highest degree is $5\rightarrow$ $a^{2}b^{3}= 2+3= 5$

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Example Question #1 : How To Find The Degree Of A Polynomial

What is the degree of the polynomial?

12x^2y^3+6xy^4z-2xz+1

Possible Answers:

Correct answer:

Explanation:

To find the degree of the polynomial, add up the exponents of each term and select the highest sum.

12x²y³: 2 + 3 = 5

6xy⁴z: 1 + 4 + 1 = 6

2xz: 1 + 1 = 2

The degree is therefore 6.

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Example Question #2 : Polynomial Functions

Let $\small f(x)=x^2-10$ , $\small g(x)=x^3$ , and $\small h(x)=2x$ . What is $\small f(h(g(x)))$ ?

Possible Answers:

$\small 6x^2-60$

$\small 8x^6-240x^4+960x^2-8000$

$\small \small 2x^6-10$

$\small 64x^6-10$

$\small 4x^6-10$

Correct answer:

$\small 4x^6-10$

Explanation:

When solving functions within functions, we begin with the innermost function and work our way outwards. Therefore:

$\small h(g(x))=2(x^3)=2x^3$

and

$\small f(h(g(x)))=(2x^3)^2-10=4x^6-10$

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Example Question #1 : Polynomial Functions

Let $\small f(x)=\sqrt[3]{x^2}$ , $\small g(x)=x^3$ , and $\small h(x)=\sqrt[4]{x}$ . What is $\small f(g(h(x)))$ ?

Possible Answers:

$\small \sqrt{x}$

$\small \sqrt[3]{x}$

$\small \sqrt[4]{x^3}$

$\small \sqrt[4]{x}$

$\small \sqrt[3]{x^2}$

Correct answer:

$\small \sqrt{x}$

Explanation:

This problem relies on our knowledge of a radical expression $\small \sqrt[b]{x^a}$ equal to $\small x^\frac{a}{b}$ . The functions are subbed into one another in order from most inner to most outer function.

$\small g(h(x))=(\sqrt[4]{x})^3=x^\frac{3}{4}$

and

$\small f(g(h(x)))=\sqrt[3]{(x^\frac{3}{4})^2}=\sqrt[3]{x^\frac{6}{4}}=x^\frac{6}{12}=x^\frac{1}{2}=\sqrt{x}$

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Example Question #1 : Polynomial Functions

Evaluate $\small f(g(9))$ if $\small f(x) = x^2 + 10$ and $\small g(x) = \sqrt{x^2 -5x+13}$

Possible Answers:

7839

$\textup{Undefined}$

Correct answer:

Explanation:

In problems with functions within one another, we must first solve the innermost function and then proceed outwards. Therefore, the first step is solving $\small g(9)$ :

$\small g(9) =\small \sqrt{(9)^2 - 5(9)+13}=\sqrt{81-45+13}=\sqrt{49}=\pm7$

Now, we must find the values of $\small f(\pm 7)$ :

$\small f(\pm 7)= (\pm 7)^2 +10=49+10=59$

Because our x term is squared in this function, both values end up being the same. Therefore, 59 is our final answer.

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Example Question #1 : Polynomial Functions

Evaluate $\small f(g(-9))$ if $\small \small f(x)=\sqrt{x^2-4}$ and $\small g(x)=-\frac{1}{3}x - 2$

Possible Answers:

$\textup{Undefined}$

$\small -\frac{38}{3}$

Correct answer:

$\textup{Undefined}$

Explanation:

Beginning with the innermost function, we must first solve for $\small \small g(-9)$ :

$\small g(9)=-\frac{1}{3}(-9)-2=3-2=1$

We then take this value and plug it into $\small f(x)$ :

$\small f(1)=\sqrt{(1)^2-4}=\sqrt{-3}$

This has no value in the real number plane, and the answer is therefore undefined.

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Example Question #2 : Polynomial Functions

g(x)=f(-x) and $f(x)=-3x^{2}-7x+4$ .

Determine g(x) .

Possible Answers:

$3x^{3}+7x^{2}-4x$

$-3x^{2}+7x+4$

$-3x^{2}+7x-4$

$3x^{2}+7x+4$

$-3x^{2}-7x+4$

Correct answer:

$-3x^{2}+7x+4$

Explanation:

Substituting -x into f(x). This has no effect on the 1st and 3rd terms. This changes the sign of the middle term.

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Example Question #73 : How To Find The Degree Of A Polynomial

Which of the following depicts a polynomial in standard form?

Possible Answers:

None of the other answers are correct.

$\small x^4+x^2+6x-2$

$\small x^3+2x^2+3x^4+5x+6$

$\small 6+5x+3x^4+2x^2+x^3$

$\small 2+6x+x^2+x^4$

Correct answer:

$\small x^4+x^2+6x-2$

Explanation:

A polynomial in standard form is written in descending order of the power. The highest power should be first, and the lowest power should be last.

$\small \small x^4+x^2+6x-2$

The answer has the powers decreasing from four, to two, to one, to zero.

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Example Question #3 : Polynomial Functions

A polynomial consists of one or more terms where each tem has a coefficient and one or more variables raised to a whole number exponent. A term with an exponent of 0 is a constant.

Identify the expression below that is not a polynomial:

$-2x^{2} +3x$
$3x^{4}-4xy^{2} -10x+5$
$-3x^{3} + 2x -3\sqrt{x}-5$

Possible Answers:

Correct answer:

Explanation:

Expression 5 has the term $-3\sqrt{x}$ , which violates the definition of a polynomial. The exponent must be a whole number.

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Algebra II : Polynomial Functions

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : How To Find The Degree Of A Polynomial

Example Question #1 : How To Find The Degree Of A Polynomial

Example Question #1 : How To Find The Degree Of A Polynomial

Example Question #2 : Polynomial Functions

Example Question #1 : Polynomial Functions

Example Question #1 : Polynomial Functions

Example Question #1 : Polynomial Functions

Example Question #2 : Polynomial Functions

Example Question #73 : How To Find The Degree Of A Polynomial

Example Question #3 : Polynomial Functions

All Algebra II Resources

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