Exercise 178 Page 433

a Polynomials are expressions that can be written as a sum of terms of the form.

(any number)* x^((whole number)) Let's rewrite some of the variables and the constant, and check if it follows the description of a polynomial. ( any number)* x^(( whole number)) ↓ f(x)= 3x^3- 2x^2+ 5x^0 As we can see, the equation is a polynomial. The highest power is 3, which means the polynomial is of the third degree.

b Again, lets check if the equation fits the description of a polynomial.

( any number)* x^(whole number) ↓ y= 0.25x^7- 5x^1 The equation is a polynomial. The highest power is 7, which means the polynomial is of the seventh degree.

c Examining the equation, we notice that it contains a term that does not follow the description of a polynomial.

( any number)* x^(whole number) ↓ y= 3^x+ 1x^2 The equation is not a polynomial.

d Notice that the square root of a number can be rewritten as the number raised to the power of 0.5. With this information, we see that the function is not a polynomial.

( any number)* x^(whole number) ↓ f(x)= 1x^2- 1x^(0.5)+ 2x^0 Since 0.5 is not a whole number, the expression is not a polynomial.

e Before we analyze the expression, we will begin by simplifying the equation.

Q(x)=3(x-4)^2(x+2)

▼

Simplify right-hand side

ExpandPosPerfectSquare

(a+b)^2=a^2+2ab+b^2

Q(x)=3(x^2-8x+16)(x+2)

MultPar

Multiply parentheses

Q(x)=3(x^3-8x^2+16x+2x^2-16x+32)

AddSubTerms

Add and subtract terms

Q(x)=3(x^3-6x^2+32)

Distr

Distribute 3

Q(x)=3x^3-18x^2+96

Now we can analyze the expression. ( any number)* x^(whole number) ↓ Q(x)= 3x^3- 18x^2+ 96x^0 The equation is a polynomial. The highest power is 3, which means the polynomial is of the third degree.

f Notice that an expression on the form 1a^b can be rewritten as a^(-b). With this information, we can rewrite the equation and determine if it is a polynomial.

( any number)* x^(whole number) ↓ y= 1x^2- 3x^1+ 5x^0- 2(x-2)^(- 1) The equation is not a polynomial.