Polynomial Functions Identification

Polynomial Functions

A polynomial function is defined as a function that is a sum or difference of monomials. This means it consists of terms where each term is a real number, a variable, or the product of a real number and variables with non-negative integer exponents or powers.

Example 1

Given: f(x)=−2x(4x2)+4f(x) = -2x(4x^2) + 4

First, simplify the expression:

−2x∗4x2=−8x3-2x * 4x^2 = -8x^3

So, f(x)=−8x3+4f(x) = -8x^3 + 4

Here, f(x)f(x) is a polynomial function because it is a sum of two monomials:

  • −8x3-8x^3 is a product of a real number (-8) and the variable xx raised to the third power, where 3 is a non-negative integer.

  • 44 is a real number.

Therefore, f(x)f(x) satisfies the definition of a polynomial function.

Example 2

Given: g(x)=x2(5x3−2)g(x) = x^2(5x^3 - 2)

First, distribute x2x^2:

x2∗5x3=5x5x^2 * 5x^3 = 5x^5
x2∗−2=−2x2x^2 * -2 = -2x^2

So, g(x)=5x5−2x2g(x) = 5x^5 - 2x^2

g(x)g(x) is a difference of two monomials. Both terms are a product of a real number and the variable xx raised to a non-negative integer power. Thus, g(x)g(x) is a polynomial function.

Example 3

Given: h(x)=4x+9h(x) = 4\sqrt{x} + 9

Rewrite the square root of xx using a rational exponent:

x=x12\sqrt{x} = x^{\frac{1}{2}}

So, h(x)=4x12+9h(x) = 4x^{\frac{1}{2}} + 9

While 99 is a monomial, 4x124x^{\frac{1}{2}} is not. The term 4x124x^{\frac{1}{2}} is not a monomial because, although 4 is a real number, the exponent on the variable xx is 12\frac{1}{2}, and 12\frac{1}{2} is not a non-negative integer. Therefore, h(x)h(x) is not a polynomial function.

Example 4

Given: j(x)=7x+3j(x) = \frac{7}{x} + 3

Rewrite 7x\frac{7}{x} using a negative exponent:

7x=7x−1\frac{7}{x} = 7x^{-1}

So, j(x)=7x−1+3j(x) = 7x^{-1} + 3

The first term, 7x−17x^{-1}, is not a monomial because the exponent on the variable xx is −1-1, which is not a non-negative integer. Thus, j(x)j(x) is not a polynomial function because it is not a sum or difference of monomials.

A polynomial function can be visualized like building blocks made of numbers and variables (x, x^2, x^3, etc.). Each block (term) is either a number, a variable, or a number multiplied by variables with positive whole number powers.

Example 1: Visualizing f(x)=−2x(4x2)+4f(x) = -2x(4x^2) + 4

  1. Simplify: Imagine combining blocks. f(x)=−8x3+4f(x) = -8x^3 + 4. So, we have a block of −8x3-8x^3 and a block of 44.

  2. Check the blocks: The −8x3-8x^3 block has −8-8 (a number) and x3x^3 (x to the power of 3). The 44 block is just a number.

  3. Polynomial? Since both blocks fit our description (numbers and variables with positive whole number powers), it’s a polynomial function.

Example 2: Visualizing g(x)=x2(5x3−2)g(x) = x^2(5x^3 - 2)

  1. Distribute: Think of spreading x2x^2 across (5x3−2)(5x^3 - 2). This gives us g(x)=5x5−2x2g(x) = 5x^5 - 2x^2.

  2. Check the blocks: We have 5x55x^5 and −2x2-2x^2. Both are numbers multiplied by x to a positive whole number power.

  3. Polynomial? Yes, because each block follows the rule.

Example 3: Visualizing h(x)=4x+9h(x) = 4\sqrt{x} + 9

  1. Rewrite: Remember that x\sqrt{x} is the same as x12x^{\frac{1}{2}}. So, h(x)=4x12+9h(x) = 4x^{\frac{1}{2}} + 9.

  2. Check the blocks: We have 4x124x^{\frac{1}{2}} and 99. But wait! The power )frac12)frac{1}{2} is not a whole number.

  3. Polynomial? No, because one of the blocks has x to a non-whole number power.

Example 4: Visualizing j(x)=7x+3j(x) = \frac{7}{x} + 3

  1. Rewrite: Remember that )frac7x)frac{7}{x} is the same as 7x−17x^{-1}. So, j(x)=7x−1+3j(x) = 7x^{-1} + 3.

  2. Check the blocks: We have 7x−17x^{-1} and 33. The power −1-1 is a negative number, not a positive whole number.

  3. Polynomial? No, because one of the blocks has x to a negative power.