Question

1. Explain why the expression 1331x^3-125y^3 is called a difference of two cubes. State the general expression for the factored form of a difference of two cubes. Factor the given expression.

2. Prove that (2x+3y)(4x^2-6xy+9y^2) + (2x-3y)(4x^2+6xy+9y^2)/16x^3 is equivalent to 1 for x^1 0.

3. The polynomial -x3-vx2+2x+w has a remainder of 4 when divided by x+2 and a remainder of 119 when divided by x-3. What are the values of v and w? (v & w eR)

4. The volume of a box is V(x)=x^3-15x^2+66x-80. Find expressions for the dimensions of the box in terms x.

5. a).Calculate (x^5+9x^4-3x^3+5x+8) / (x^2+3x+1) using long division.
b).Write a division statement.

6.Calculate (-2x^4 +x^3-8x^2+11) / (x-6) using synthetic division.


Please show all the work. Please and thank you.

Answer 1

1331 = 11^3
125 = 5^3
so you have
(11 x)^3 - (5 y)^3

a^3 - b^3 = (a-b)(a^2+ab+b^2)
so
(11x-5y)(121x^2+ 55xy + 25y^2)

.... and no, I am not going to do them all.

Answer 2

1.

11³ = 1331 , 5³ = 125

1331 x³ - 125 y³ = ( 11 x )³ - ( 5 y )³

a³ - b³ = ( a - b ) ( a² + a ∙ b + b² )

so

( 11 x )³ - ( 5 y )³ = ( 11 x - 5 y ) [ ( 11 x )² + 11 x ∙ 5 y + ( 5 y )² ] =

( 11 x - 5 y ) ( 121 x ² + 55 x y + 25 y² )

2.

a³ + b³ = ( a + b ) ( a² - a ∙ b + b² )

so

( 2 x + 3 y ) ( 4 x² - 6 x y + 9 y² ) =

( 2 x + 3 y ) [ ( 2 x )² - 2 x ∙ 3 y + ( 3 y )² ] = ( 2 x )³ + ( 3 y )³

a³ - b³ = ( a - b ) ( a² + a ∙ b + b² )

so

( 2 x - 3 y ) ( 4 x² + 6 x y + 9 y² ) =

( 2 x - 3 y ) [ ( 2 x )² + 2 x ∙ 3 y + ( 3 y )² ] = ( 2 x )³ - ( 3 y )³

[ ( 2 x + 3 y ) ( 4 x² - 6 x y + 9 y² ) + ( 2 x - 3 y ) ( 4 x²+ 6 x y + 9 y² ) ]/16 x³ =

[ ( 2 x )³ + ( 3 y )³ + ( 2 x )³ - ( 3 y )³ ] / ( 16 x³ ) = [ ( 2 x )³ + ( 2 x )³ ] / 16 x³ =

( 8 x³ + 8 x³ ) / 16 x³ = 16 x³ / 16 x³ = 1

3.

- x³ - v x² + 2 x + w

Remainder theorem:

When p(x) is divided by x - c the remainder is p(c)

In this case:

for

c = - 2

x - c = x - ( - 2 ) = x + 2

Remainder:

p(-2)= - ( - 2 )³ - v ∙ ( - 2 )² + 2 ∙ ( - 2 ) + w = w - 4 v + 4 = 4

for

c = 3

x - c = x - 3

p(3)= - ( 3 )³ - v ∙ 3² + 2 ∙ 3 + w = w - 9 v - 21 = 119

Now you must solve:

w - 4 v + 4 = 4

Suptract 4 to both sides

w - 4 v = 0

w - 9 v - 21 = 119

Add 21 to both sides

w - 9 v = 140

So system of equations is:

w - 4 v = 0

w - 9 v = 140
__________

The solution is:

v = - 28 , w = - 112

Your polinom:

- x³ - v x² + 2 x + w = - x³ + 28 x² + 2 x -112

4.

In google paste:

factoring polynomials calculator - eMathHelp

When you see list of results click on:

Factoring Polynomials Calculator - eMathHelp

When page be open in rectangle Enter a polynomial paste:

x^3-15x^2+66x-80

and click option CALCULATE

You will see solution step-by-step

5.

In google paste:

polynomial long division calculator - eMathHelp

When you see list of results click on:

Polynomial Long Division Calculator - eMathHelp

When page be open in rectangle Divide (dividend) paste:

x^5+9x^4-3x^3+5x+8

in rectangle By (divisor) paste:

x^2+3x+1

and click option CALCULATE

You will see solution step-by-step

6.

In google paste:

polynomial synthetic division calculator - eMathHelp

When you see list of results click on:

Synthetic Division Calculator - eMathHelp

When page be open in rectangle Divide (dividend) paste:

-2x^4 +x^3-8x^2+11

in rectangle By (divisor) paste:

x-6

You will see solution step-by-step