IDENTIFY POLYNOMIAL FUNCTIONS AND THEIR DEGREE

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 

Problem 11 :

Complete the statement with always, sometimes, or never. Explain your reasoning.

a) The terms of a polynomial are ________ monomials.

b) The difference of two trinomials is _________ a trinomial.

c) A binomial is ________ a polynomial of degree

d) The sum of two polynomials is _________ a polynomial.

Solution :

a) Algebraic terms will consists of numerical value and variable with power, the addition and subtraction sign will separate terms. So, 

The terms of a polynomial are always monomials.

b) Let us consider two trinomials, 2a + 3b - c and 5a - 2b + 2c

= 2a + 3b - c -(5a - 2b + 2c)

= 2a - 5a + 3b + 2b - c - 2c

= -3a + 5b - 3c

So, 

The difference of two trinomials is sometimes a trinomial.

c) A binomial is sometimes a polynomial of degree 2.

d) The sum of two polynomials is always  a polynomial.

Problem 12 :

Write 15x − x³ + 3 in standard form. Identify the degree and leading coefficient of the polynomial.

Solution :

To write the polynomial in standard form, we have to arrange the terms from greater exponent to the least exponent.

= − x³ + 15x + 3

Degree = 3

Leading coefficient = -1

Problem 13 :

Write each polynomial in standard form. Identify the degree and classify each polynomial by the number of terms.

a. −3z⁴

b. 4 + 5x² − x

c. 8q + q⁵

Solution :

a. −3z⁴

It is in standard form already.

Degree = 4

Leading coefficient = -3

b. 4 + 5x² − x

Standard form :

5x² − x + 4

Degree of the polynomial is 2

Leading coefficient is 5.

c. 8q + q⁵

Standard form :

q⁵ + 8q

Degree of the polynomial is 5

Leading coefficient is 1.

Problem 14 :

Write two binomials that have the product x² − 121. Explain.

Solution :

x² − 121 = x² − 11²

= (x - 11)(x + 11)

So, the required binomials are (x - 11)(x + 11)

Problem 15 :

The cost (in dollars) of making b bracelets is represented by 4 + 5b. The cost (in dollars) of making b necklaces is represented by 8b + 6. Write a polynomial that represents how much more it costs to make b necklaces than b bracelets.

Solution :

Cost of making bracelets = 4 + 5b

Cost of making necklaces = 8b + 6

Difference between = 8b + 6 - (4+ 5b)

= 8b + 6 - 4 - 5b

= 8b - 5b + 6 - 4

= 3b + 2

Problem 16 :

Write the degree of the given polynomials :

i) (2x + 4)³

ii) (t³ + 4) (t³ + 9)

Solution :

i) (2x + 4)³

Degree of the polynomial is 3

ii) (t³ + 4) (t³ + 9)

Degree of the polynomial is 6.