Polynomial Functions

Polynomial Functions

Terms and Degrees of a Polynomial Function

Polynomial Function

• It is a function defined by a polynomial expression.

• The degree of the polynomial is the largest exponent. In the general form, the degree is n.

How will you know if an expression is a polynomial or not?

Non-examples of polynomial expressions

How can you say that an expression is not a polynomial expression?

Practice

Determine whether the given function is a polynomial function or not.

Multiple Choice

 $f\left(x\right)=x^2+4x-5$  

• Polynomial function

• Not a polynomial function

Multiple Choice

 $f\left(x\right)=\frac{1}{2}x^3+\frac{3}{4}x^2-5x-10$  

• Polynomial function

• Not a polynomial function

Multiple Choice

 $f\left(x\right)=\frac{5}{x+1}-9x+8$  

• Polynomial function

• Not a polynomial function

Multiple Choice

 $f\left(x\right)=\frac{x^2}{4}-\frac{3x}{2}+17$  

• Polynomial function

• Not a polynomial function

Practice

Determine the degree of the polynomial function.

Fill in the Blanks

 $p\left(x\right)=4x^3+5x^2-2x+1$  

Fill in the Blanks

 $p\left(x\right)=5-7x+x^2$  

Fill in the Blanks

 $f\left(x\right)=5\pi-5x^3+x^7-x$  

For more practice, follow the link below.

https://www.expii.com/t/find-degree-of-polynomial-product-4476

Operations on Polynomial Functions

Sum, Difference, and Product of Polynomial Functions

Let

 $f\left(x\right)=x-2$  and  $g\left(x\right)=x^2-4$  . Find the functions  $f+g,\ f-g,$  and  $fg$  

 $f+g$  

Let

 $f\left(x\right)=x-2$  and  $g\left(x\right)=x^2-4$  . Find the functions  $f+g,\ f-g,$  and  $fg$  

 $f-g$  

Let

 $f\left(x\right)=x-2$  and  $g\left(x\right)=x^2-4$  . Find the functions  $f+g,\ f-g,$  and  $fg$  

 $fg$  

Let

 $f\left(x\right)=x-2$  and  $g\left(x\right)=x^2-4$  . Find the functions  $f+g,\ f-g,$  and  $fg$  

Practice

Multiple Choice

Given

f(x) = 3x² + 7x and g(x) = 2x² - x - 1, find (f + g)(x).

• 11x² - 1

• 5x⁴ + 6x² - 1

• 5x² + 6x - 1

• 5x² + 8x - 1

Multiple Choice

 $f(x)=x³-3x$  
 $g(x)=-4x+1$  
 $Find\ fg(x)$  

• -4x³-5x²-3

• -4x³+2x²-3x

• -4x⁴+x³+12x²-3x

• -4x⁴-x³+12x²⁺3x

Multiple Choice

f(x) = 4x+8

g(x) = x+3,

Find fg(x)

• 4x²+24

• 4x²+20x+24

• 4x²+12x+24

• 4x²+4x+24

Multiple Choice

f(x)=x²+1

g(x)=2x-5

Find (f-g)(x)

• x²-2x+6

• -x²+2x+4

• -x²-2x-6

• x²-2x-4

Multiple Choice

f(x) = 9-3x

g(x) = 5x-7

Find f(x)+g(x).

• 14x-10

• -8x+2

• 2x+2

• 2x-2

Multiple Choice

f(n)=2n+1

g(n)=3n

Find f(n)+g(n)

• 5n+1

• -5n+1

• -2n+1

• -6n-1

Multiple Choice

f(x) = x²+9

g(x) = x-9

Find f(x)-g(x).

• x²+x+9

• x²-x+9

• x²-x

• x²-x+18

Dividing Polynomials by Long Division

https://www.youtube.com/watch?v=l6_ghhd7kwQ

 Long Division of Polynomials by PatricKJMT

Practice

Long Division

Multiple Choice

 $(3x^3+5x-1)÷(x+1)$  

•  $3x^3-3x^2+8x-9$  

•  $3x^2-3x+8-\frac{9}{x+1}$  
  

•  $3x^2+3x+8+\frac{7}{x+1}$  
  

•  $3x^3+3x^2+8x+7$  
  

Multiple Choice

• $x+5+\frac{2}{2x-4}$
  

• x - 8

• $x-5+\frac{3}{2x-4}$
  

• x + 5

Multiple Choice

 $(8p^3-38p^2-17p+43)÷(p-5)$  

•  $8p^2+2p-7\ +\frac{8}{p-5}$  

•  $8p^2+p-9+\frac{9}{p-5}$  
  

•  $8p^2+p-5+\frac{13}{p-5}$  
  

•  $8p2+p-10+\frac{4}{p-5}$  
  

Synthetic Division

https://www.youtube.com/watch?v=AWCQBLthbNI

Synthetic Division... How? by NancyPi

Practice

Synthetic Division

Multiple Choice

For synthetic division, what would be the number in the left hand box?

• 0
• 1
• 2
• 3

Multiple Choice

If you were dividing x⁶ + 4x³ + 2, how many 0's would you need when setting up the top row of your synthetic division?

• 0
• 2
• 4
• 6

Multiple Choice

Which term is missing in this problem?
2x³ + 5x² + 9 ÷
x + 3

• x⁴
• x³
• x²
• x

Multiple Choice

• 2x³ + 3x + (7/x-4)
• 2x² + 3x + 8 + (7/x-4)
• 2x² + 3x + 8
• 2x² - x + 10 + (6/x-4)

Multiple Choice

• x - 7
• x² - 7
• x + 7
• x² + 3x - 54