IDENTIFY POLYNOMIAL FUNCTIONS AND THEIR DEGREE
A polynomial function is a function that involves only non-negative integer powers or only positive integer exponents of a variable in an equation like the quadratic equation, cubic equation.
What is degree of the polynomial ?
A polynomial’s degree is the highest or the greatest power of a variable in a polynomial. The degree indicates the highest exponential power in the polynomial.
Find degree of the polynomial consists of one variable :
For example, considering the powers of the polynomial
3x2 + 2x
2 is the highest exponent. So, 2 is the degree of the polynomial.
Find degree of the polynomial consists of multi variable :
For example, considering the powers of the polynomial
3x2y + 2x3y2 + 3xy
Degree of the first term 3x2y = 2 + 1 ==> 3
Degree of the second term 2x3y2 = 3 + 2 ==> 5
Degree of the third term 3xy = 1 + 1 ==> 2
The highest is 5. So, the degree of the polynomial is 5.
Which of the following are polynomial functions? For those that are polynomial functions, state the degree and leading coefficient. For those that are not, explain why not.
Problem 1 :
f(x) = 4x³ - 5x – 1/2
Solution :
Degree = 3
Leading coefficient = 4
The function is a polynomial function.
Problem 2 :
g(x) = 6x-4 + 7
Solution :
The function is not a polynomial function because the terms 6x-4 has a negative exponent.
Problem 3 :
h(x) = √(9x4 + 16x²)
Solution :
The function is not a polynomial function because it has rational exponent.
Problem 4 :
k(x) = 15x – 2x4
Solution :
The function is a polynomial function.
Degree = 4
Leading coefficient = -2
Problem 5 :
f(x) = 3x-5 + 17
Solution :
The function is not a polynomial function because it has negative exponent.
Problem 6 :
f(x) = -9 + 2x
Solution :
The function is a polynomial function.
Degree = 1
Leading coefficient = 2
Problem 7 :
f(x) = 2x5 – 1/2x + 9
Solution :
The function is a polynomial function.
Degree = 5
Leading coefficient = 2
Problem 8 :
f(x) = 13
Solution :
The function is a polynomial function.
Degree = 0
Leading coefficient = 13
Problem 9 :
h(x) = (27x³ + 8x6)
Solution :
The function is not a polynomial function because of the cube root.
Problem 10 :
k(x) = 4x – 5x²
Solution :
The function is a polynomial function.
Degree = 2
Leading coefficient = -5
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