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.
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
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.
1. The expression 1331x^3-125y^3 is called a difference of two cubes because it can be written as (11x)^3 - (5y)^3. In general, the factored form of a difference of two cubes is given by:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
Now let's factor the given expression:
1331x^3 - 125y^3 = (11x)^3 - (5y)^3
= (11x - 5y)((11x)^2 + (11x)(5y) + (5y)^2)
= (11x - 5y)(121x^2 + 55xy + 25y^2)
2. To prove that (2x+3y)(4x^2-6xy+9y^2) + (2x-3y)(4x^2+6xy+9y^2)/16x^3 is equivalent to 1 for x ≠ 0, we can substitute x =/= 0 into the expression and simplify:
[(2x + 3y)(4x^2 - 6xy + 9y^2) + (2x - 3y)(4x^2 + 6xy + 9y^2)] / (16x^3)
= [(2x)(4x^2 - 6xy + 9y^2) + (3y)(4x^2 - 6xy + 9y^2) + (2x)(4x^2 + 6xy + 9y^2) - (3y)(4x^2 + 6xy + 9y^2)] / (16x^3)
= [(8x^3 - 12x^2y + 18xy^2) + (12xy^2 - 18y^3) + (8x^3 + 12x^2y + 18xy^2) - (12xy^2 + 18y^3)] / (16x^3)
= [16x^3] / (16x^3)
= 1 (for x ≠ 0)
3. To find the values of v and w in the polynomial -x^3 - vx^2 + 2x + w given the remainders when divided by x+2 and x-3, we can use the Remainder Theorem.
When divided by x+2, we have a remainder of 4, so we set up the equation:
-(-2)^3 - v(-2)^2 + 2(-2) + w = 4
-8 - 4v - 4 + w = 4
-12 - 4v + w = 4
w - 4v = 16
When divided by x-3, we have a remainder of 119, so we set up the equation:
-(-3)^3 - v(-3)^2 + 2(-3) + w = 119
-27 - 9v - 6 + w = 119
-33 - 9v + w = 119
w - 9v = 152
Solving the system of equations w - 4v = 16 and w - 9v = 152, we get:
Subtracting the two equations, we have: (w - 9v) - (w - 4v) = 152 - 16
-5v = 136
v = -27.2
Substituting v = -27.2 into w - 4v = 16, we get:
w - 4(-27.2) = 16
w + 108.8 = 16
w = -92.8
So, the values of v and w are v = -27.2 and w = -92.8.
4. The volume of the box is given by V(x) = x^3 - 15x^2 + 66x - 80. To find expressions for the dimensions of the box in terms of x, we can find the roots of the cubic equation.
Setting V(x) = 0, we have:
x^3 - 15x^2 + 66x - 80 = 0
Using synthetic division or a numerical technique like Newton's method, we find that x = 4, x = 5, and x = 8 are the roots of the equation.
Therefore, the dimensions of the box in terms of x are:
Length = x - 4
Width = x - 5
Height = x - 8
5. a) To calculate (x^5 + 9x^4 - 3x^3 + 5x + 8) / (x^2 + 3x + 1) using long division:
___________________________________________
(x^2 + 3x + 1) | (x^5 + 9x^4 - 3x^3 + 0x^2 + 5x + 8)
- (x^5 + 3x^4 + x^2)
______________________
6x^4 - 3x^3 + x^2 + 5x + 8
- (6x^4 + 18x^3 + 6x)
______________________
-21x^3 + (x^2 - x + 5x + 8)
- (-21x^3 - 63x^2 - 21x)
______________________
(64x^2 + 26x + 8)
So, (x^5 + 9x^4 - 3x^3 + 5x + 8) / (x^2 + 3x + 1) = x^3 - x^2 + 5x + (64x^2 + 26x + 8) / (x^2 + 3x + 1)
b) The division statement is:
(x^5 + 9x^4 - 3x^3 + 5x + 8) / (x^2 + 3x + 1) = x^3 - x^2 + 5x + (64x^2 + 26x + 8) / (x^2 + 3x + 1)
6. To calculate (-2x^4 + x^3 - 8x^2 + 11) / (x - 6) using synthetic division:
6 | -2 1 -8 0 11
0 -12 42 252
___________________
-2 1 -20 42 263
So, (-2x^4 + x^3 - 8x^2 + 11) / (x - 6) = -2x^3 + x^2 - 20x + 42 with a remainder of 263.