How do you factor this polynomial? x^3 -4x^2 -25x +100?

Question Number 1

For this polynomial equation x^3 -4*x^2 -25*x +100 = 0, answer the following questions :

A. Solve by Factorization

Answer For Question 1

x^3 -4*x^2 -25*x +100 = 0

And we get P(x)=x^3 -4*x^2 -25*x +100

Now, we can look for the roots of P(x) using various Algorithm :

1A. Solve by Factorization

x^3 -4*x^2 -25*x +100 = 0

Separate : ( x^3 -4*x^2 ) + ( -25*x +100 ) = 0

Commutative Law : ( x^3 -25*x ) + ( -4*x^2 +100 ) = 0

Distributive Law : x*( x^2 -25 ) + -4*( x^2 -25 ) = 0

Factor : ( x -4 )*( x^2 -25 ) = 0

Separate : ( x -4 )*( ( x^2 -5*x ) + ( 5*x -25 ) ) = 0

Commutative Law : ( x -4 )*( ( x^2 +5*x ) + ( -5*x -25 ) ) = 0

Distributive Law : ( x -4 )*( x*( x +5 ) + -5*( x +5 ) ) = 0

Factor : ( x -4 )*( x -5 )*( x +5 )

So the Polynomial have 3 roots :

x1 = 4

x2 = 5

x3 = -5

x^3 -4x^2 -25x +100= 0

Factor by grouping

(x^3 -4x^2) -(25x - 100) = 0

x^2(x-4)-25(x-4) = 0

(x^2-25)(x-4) = 0

Now factor x^2 -25 as a difference of squares

(x-5)(x+5)(x-4) = 0

x^3 -4x^2 -25x +100 =

= x^2(x - 4) - 25(x - 4) =

= (x - 4)(x^2 - 25) =

= (x - 4)(x + 5)(x - 5)

x-intercepts are

x = 4, x = 5, x = -5

x^3 - 4x^2 - 25x + 100

x^2(x-4) - 25(x-4)

(x^2-25)(x-4) = 0

The x-intercepts are therefore (+/-5,0) and (4,0).

x^3 – 4x² – 25x + 100 =∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ factor gcf out of 1st 2 and last 2 terms

x²(x – 4) – 25(x – 4) =∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙and then factor out the common binomial

(x² – 25)(x – 4) = ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ and recognize 1st binomial is difference of squares

(x + 5)(x – 5)(x – 4)