Multiply and simplify please

Question

Multiply and Simplify

 

x^2-2x-48  *  9x^3-4x

3x^3-2x^2     8x-64

 

Answer 1

Multiplying the fractions results in:

f(x)=(x^2-2x-48)(9x^3-4x)/((3x^3-2x^2)(8x-64))

f(x)=x(x-8)(x+6)(9x^2-4)/(8x^2(3x-2)(x-8))

9x^2-4 = (3x)^2-2^2 = (3x+2)(3x–2)

f(x)=x(x-8)(x+6)(3x+2)(3x–2)/(8x^2(3x-2)(x-8))

f(x)=x(x-8)(3x–2)/(x(x-8)(3x–2))
* (x+6)(3x+2)/(8x)

x(x-8)(3x–2)/(x(x-8)(3x–2))
= {1 if x ≠ 0, 2/3, or 8; UNDEFINED otherwise }
I.e., there are “Holes” at x = 0, 2/3, or 8.

f(x)=(x+6)(3x+2)/(8x), x ≠ 0, 2/3, or 8

 

Since x is in the denominator of this reduced form there will be a vertical asymptote at x = 0.

 

f(x)=(3x^2+20x+12)/(8x), x ≠ 0, 2/3, or 8

 

Do long division of numerator by denominator:

 

3x^2+20x+12 | 3x/8

3x^2

––––––––––––

20x+12 | 20/8

20x

––––––

12

 

f(x)=3x/8 + 20/8 + 12/(8x), x ≠ 0, 2/3, or 8

 

f(x)=3x/8 + 5/2 + 3/(2x), x ≠ 0, 2/3, or 8

 

This is the simplest form; it shows a slant asymptote of y=3x/8 + 5/2.

Answer 2

Hi Theresa,

 

x²-2x-48=x²-8x+6x-48

 

              =x(x-8)+6(x-8)

             =(x+6)(x-8)

 

3x³-2x²=x²(3x-2)

 

9x³-4x=x(9x²-4)

 

          =x(3x-2)(3x+2)

 

8x-64=8(x-8)

 

so the given expression becomes,

 

(x+6)(x-8) × (x)(3x-2)(3x+2)

x²(3x-2)         8(x-8)

 

(x+6)(3x+2)

    8x

 

For x=-6,-2/3 and 0 the solution is not defined.

Answer 3

Hi Theresa,

 

factors of

x^2-2x-48=(x-8)(x+6)

3x^3-2x^2 = x^2(3x-2)

9x^3-4x = x(9x^2-4)= x(3x-2)(3x+2)

8x-64 = 8( x-8)

 

using all this factors,

(x-8)(x+6)  *  x(3x-2)(3x+2)

________      _____________

x^2(3x-2)      8(x-8)

 

=(x+6)(3x+2)

  __________

     8 x

 

=3x^2+20x+12

  ___________  

     8x                                     

 

 

Meena from Strongsville,OH