Compute Difference Quotient

[nycmath]

Precalculus
David Cohen
Edition 3
Chapter 3, Srction 3.1

Compute the difference quotient for the function.

See attachment.

[Eigenvalue]

51. [tex]\frac{4x²-4a²}{x-a}[/tex]
=4([tex]\frac{(x+a)(x-a)}{x-a}[/tex])
=4(x+a)=4x+4a

[Eigenvalue]

52. [tex]\frac{-2x²+9-(-2a²+9)}{x-a}[/tex]
=[tex]\frac{-2x²+9+2a²-9}{x-a}[/tex]
=[tex]\frac{-2x²+2a²}{x-a}[/tex]
=-2([tex]\frac{x²-a²}{x-a}[/tex])
=-2(x+a)=-2x-2a

[Eigenvalue]

53. [tex]\frac{x²-2x+4-a²+2a-4}{x-a}[/tex]
=[tex]\frac{x²-a²-2x+2a}{x-a}[/tex]
=[tex]\frac{(x-a)(x+a-2)}{(x-a)}[/tex]
=x+a-2

[Eigenvalue]

[tex]\frac{2x²-x+1-2a²+a-1}{x-a}[/tex]
=[tex]\frac{2x²-2a²-x+a}{x-a}[/tex]
=[tex]\frac{(x-a)(2x+2a-1)}{x-a}[/tex]
=2x+2a-1

[Eigenvalue]

55. [tex]\frac{4x³-4a³}{x-a}[/tex]
=4([tex]\frac{x³-a³}{x-a}[/tex])
=4([tex]\frac{(x-a)(x²+x+1)}{x-a}[/tex]
=4(x²+x+1)=-4x²-4x+4

[Eigenvalue]

56. [tex]\frac{-x³+2+a³-2}{x-a}[/tex]
=-([tex]\frac{x³-a³}{x-a}[/tex])
=-(x²+x+1)
=-x²-x-1

[Eigenvalue]

57.[tex]\frac{(5/(x+h)²)-(5/x²)}{h}[/tex]
=[tex]\frac{5x²-5(x+h)²}{h(x²)(x+h)²}[/tex]
=[tex]\frac{-10hx-5h²}{h(x²)(x+h)²}[/tex]
=[tex]\frac{-10x-5h}{x²(x+h)²}[/tex]

[Eigenvalue]

[tex]\frac{(-3/(x+h)²)-(-3/x²)}{h}[/tex]

=[tex]\frac{-3x²+3(x+h)²}{h(x²)(x+h)²}[/tex]
=[tex]\frac{6xh+3h²}{h(x²)(x+h)²}[/tex]
=[tex]\frac{6x+3h}{x²(x+h)²}[/tex]

[nycmath]

51. [tex]\frac{4x²-4a²}{x-a}[/tex]
=4([tex]\frac{(x+a)(x-a)}{x-a}[/tex])
=4(x+a)=4x+4a

The site only allows three thumbs up per every few hours. A job well-done. You are amazingly smart.

[nycmath]

52. [tex]\frac{-2x²+9-(-2a²+9)}{x-a}[/tex]
=[tex]\frac{-2x²+9+2a²-9}{x-a}[/tex]
=[tex]\frac{-2x²+2a²}{x-a}[/tex]
=-2([tex]\frac{x²-a²}{x-a}[/tex])
=-2(x+a)=-2x-2a

Easy to follow steps. Great work.

[nycmath]

53. [tex]\frac{x²-2x+4-a²+2a-4}{x-a}[/tex]
=[tex]\frac{x²-a²-2x+2a}{x-a}[/tex]
=[tex]\frac{(x-a)(x+a-2)}{(x-a)}[/tex]
=x+a-2

You outshine them all.

[nycmath]

[tex]\frac{2x²-x+1-2a²+a-1}{x-a}[/tex]
=[tex]\frac{2x²-2a²-x+a}{x-a}[/tex]
=[tex]\frac{(x-a)(2x+2a-1)}{x-a}[/tex]
=2x+2a-1

Terrific. Wonderful. Great work.

[nycmath]

55. [tex]\frac{4x³-4a³}{x-a}[/tex]
=4([tex]\frac{x³-a³}{x-a}[/tex])
=4([tex]\frac{(x-a)(x²+x+1)}{x-a}[/tex]
=4(x²+x+1)=-4x²-4x+4

You are cooking with gas.

[nycmath]

56. [tex]\frac{-x³+2+a³-2}{x-a}[/tex]
=-([tex]\frac{x³-a³}{x-a}[/tex])
=-(x²+x+1)
=-x²-x-1

Simple and direct. Thank you.

[nycmath]

57.[tex]\frac{(5/(x+h)²)-(5/x²)}{h}[/tex]
=[tex]\frac{5x²-5(x+h)²}{h(x²)(x+h)²}[/tex]
=[tex]\frac{-10hx-5h²}{h(x²)(x+h)²}[/tex]
=[tex]\frac{-10x-5h}{x²(x+h)²}[/tex]

Impressive and direct.

[nycmath]

[tex]\frac{(-3/(x+h)²)-(-3/x²)}{h}[/tex]

=[tex]\frac{-3x²+3(x+h)²}{h(x²)(x+h)²}[/tex]
=[tex]\frac{6xh+3h²}{h(x²)(x+h)²}[/tex]
=[tex]\frac{6x+3h}{x²(x+h)²}[/tex]

Wonderfully-done. Great math skills for sure.