Question

ПОмогите решить 2 задания. разложите на множители: а)12а^2b-27b^3; b)-40x^3-120x^2y-90xy^2; в)2a^4-16ab^3; г)([^3+8)-(3x+6). Решите уравнение x^3-x^2-25x+25=0 помогите умоля .зарание спасибо

Answer 1

1. а) 3b (4a^2 - 9b^2) = 3b (2a - 3b) (2a +3b)

     b) -10x (4x^2 + 12xy + 9y^2) = -10x (2x + 3y)^2

     c) 2a (a^3 - 8b^3) = 2a (a - 2b) (a^2 + 2ab + 4b^2)

     d) (x + 2) (x^2 - 2x + 4) - 3 (x + 2) = (x + 2) (x^2 - 2x + 4 - 3) =  (x + 2) (x^2 - 2x + 1)

 

2. (x^3 - x^2) - (25x - 25) = 0

     x^2 (x - 1) - 25 (x - 1) = 0

    (x - 1) (x^2 - 25) = 0

    (x - 1) (x - 5) (x + 5) = 0

произведение равно нулю, когда хотя бы один из множителей равен нулю, а другой приэтом не теряет смысла; тогда:

х - 1 = 0     или   х - 5 = 0    или х + 5 = 0

х = 1                     х = 5                 х =-5

Answer 2

Task 1: Factorize 12a^2b - 27b^3

To factorize the expression 12a^2b - 27b^3, we can first factor out the greatest common factor (GCF) of the terms. In this case, the GCF is 3b:

12a^2b - 27b^3 = 3b(4a^2 - 9b^2)

Next, we can apply the difference of squares formula to factor the expression inside the parentheses:

4a^2 - 9b^2 = (2a)^2 - (3b)^2 = (2a + 3b)(2a - 3b)

Therefore, the fully factorized form of 12a^2b - 27b^3 is:

3b(2a + 3b)(2a - 3b)

Task 2: Factorize -40x^3 - 120x^2y - 90xy^2

To factorize the expression -40x^3 - 120x^2y - 90xy^2, we can first factor out the greatest common factor (GCF) of the terms. In this case, the GCF is -10x:

-40x^3 - 120x^2y - 90xy^2 = -10x(4x^2 + 12xy + 9y^2)

Next, we can apply the perfect square trinomial formula to factor the expression inside the parentheses:

4x^2 + 12xy + 9y^2 = (2x + 3y)^2

Therefore, the fully factorized form of -40x^3 - 120x^2y - 90xy^2 is:

-10x(2x + 3y)^2

Task 3: Factorize 2a^4 - 16ab^3

To factorize the expression 2a^4 - 16ab^3, we can first factor out the greatest common factor (GCF) of the terms. In this case, the GCF is 2a:

2a^4 - 16ab^3 = 2a(a^3 - 8b^3)

Next, we can apply the difference of cubes formula to factor the expression inside the parentheses:

a^3 - 8b^3 = (a - 2b)(a^2 + 2ab + 4b^2)

Therefore, the fully factorized form of 2a^4 - 16ab^3 is:

2a(a - 2b)(a^2 + 2ab + 4b^2)

Task 4: Solve the equation x^3 - x^2 - 25x + 25 = 0

To solve the equation x^3 - x^2 - 25x + 25 = 0, we can use various methods such as factoring, the quadratic formula, or numerical methods. Let's try factoring first.

By observing the equation, we can see that x = 1 is a root of the equation. This means that (x - 1) is a factor of the equation. We can use polynomial long division or synthetic division to divide the equation by (x - 1):

       x^2 - 25
    -------------
x - 1 | x^3 - x^2 - 25x + 25
       - (x^3 - x^2)
       -------------
                 - 25x + 25
                 - (- 25x + 25)
                 -------------
                           0

The result of the division is x^2 - 25. Now we have factored the equation as (x - 1)(x^2 - 25) = 0.

Next, we can factor the quadratic expression x^2 - 25 using the difference of squares formula:

x^2 - 25 = (x - 5)(x + 5)

Therefore, the fully factored form of the equation x^3 - x^2 - 25x + 25 = 0 is:

(x - 1)(x - 5)(x + 5) = 0

To find the solutions, we set each factor equal to zero:

x - 1 = 0 -> x = 1 x - 5 = 0 -> x = 5 x + 5 = 0 -> x = -5

Therefore, the solutions to the equation x^3 - x^2 - 25x + 25 = 0 are:

x = 1, x = 5, x = -5