2.2 sections Polynomial Functions

Question

Please provide a video and explain how you have gotten to those answers: Today as soon as possible.

For each polynomial function find the degree, the leading term, and the leading coefficient.

1.) f(x) = 2x⁵ -5x² ---------------------> Answer: polynomial function degree: 5, the leading term: 2x⁵, the leading coefficient: 2

2.) f(x) = 2x³+7x / 3 --------------------> Answer: polynomial function degree: 3, the leading term: 2/3 x³, the leading coefficient: 2/3

3.) f(x) = nx⁴ +1 -x² -------------------> Answer: polynomial function degree: 4, the leading term: nx⁴, the leading coefficient: N

Answer 1

1.

The degree of a polynomial TERM is the exponent (or sum of exponents if there is more than one variable in a single term)

Terms are separated by plus or minus signs in front of a coefficient.

A coefficient is the number in front of a variable (or group of variables multiplied together). It tells you how many of this variable (or variable product) you have.

2x^5 has a degree of 5. Its coefficient is 2.

-5x^2 has a degree of 2. Its coefficient is -5. Note that the subtraction causes the sign of the term to be negative. It is like it was written +(-5x^2).

So, the higher degree is 5, and that is the degree of the entire polynomial. 2x^5 is the leading term.

The leading term is the term of a polynomial that has the highest degree.

The leading coefficient is the coefficient associated with the leading term.

So the leading coefficient in this case is 2.

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2. I am going to presume the polynomial function was actually written like this:

f(x) = (2x^3 +7x)/3

If you apply the distributive law,

f(x) = (2/3)x^3 + (7/3)x

Using the definitions from above, the degree of the polynomial is 3, since the leading term is (2/3)x^3. The leading coefficient is 2/3.

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3.

The term nx^4 has a degree of 4.

The term 1 is a constant and is defined to have a degree of 0. (The same value as the exponent of

x^0 = 1, if x is not equal to 0.)

The term -x^2 has a degree of 2.

So, the degree of the polynomial is 4. The leading term is nx^4. The leading coefficient is n. Yes, we can have variables that are used as constants and/or coefficients.

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If a polynomial is arranged with the degrees of each term in strictly descending order, then the polynomial is said to be in standard form. I am sure you will see this term used again (and very soon), and that is why I am bringing it up now.

The polynomials in problems 1 and 2 are in standard form; 3 is not.

If the polynomial in problem 3 were written:

nx^4 -x^2 +1

it would be in standard form.