Question

Разложите на множители 1)m^3+27n; 2)x^3-64xy^2; 3)-3a^2+18a-27; 4)2ab+10b-2a-10; 5)a^4-16.

Answer 1

1) Здесь предполагаю что выражение должно быть таким  $m^3+27n^3$. Тогда можно использовать формулу сумму кубов

$m^3+27n^3=(m+3n)(m^2-3mn+9n^2)$

2.  $x^3-64xy^2=x(x^2-64y^2)=x(x-8y)(x+8y)$

3.  $-3a^2+18a-27=-3(a^2-6a+9)=-3(a-3)^2$

4.  $2ab+10b-2a-10=2b(a+5)-2(a+5)=2(a+5)(b-1)$


5.  $a^4-16=(a^2-16)(a^2+16)=(a-4)(a+4)(a^2+16)$

Answer 2

1. m^3 + 27n:

We can rewrite the expression using the identity (a^3 + b^3) = (a + b)(a^2 - ab + b^2):

m^3 + 27n = (m)^3 + (3)^3(n) = (m + 3n)(m^2 - 3mn + 9n^2)

So the expression can be factored as (m + 3n)(m^2 - 3mn + 9n^2).

2. x^3 - 64xy^2:

We can rewrite the expression using the identity (a^3 - b^3) = (a - b)(a^2 + ab + b^2):

x^3 - 64xy^2 = (x)^3 - (4xy)^3 = (x - 4xy)(x^2 + 4xy + 16x^2y^2)

So the expression can be factored as (x - 4xy)(x^2 + 4xy + 16x^2y^2).

3. -3a^2 + 18a - 27:

We can factor out a common factor of -3:

-3a^2 + 18a - 27 = -3(a^2 - 6a + 9)

Now, we can rewrite the expression as a perfect square:

-3(a^2 - 6a + 9) = -3(a - 3)^2

So the expression can be factored as -3(a - 3)^2.

4. 2ab + 10b - 2a - 10:

We can group the terms and factor out common factors:

(2ab - 2a) + (10b - 10) = 2a(b - 1) + 10(b - 1)

Now, we can factor out the common factor (b - 1):

2a(b - 1) + 10(b - 1) = (b - 1)(2a + 10) = 2(b - 1)(a + 5)

So the expression can be factored as 2(b - 1)(a + 5).

5. a^4 - 16:

We can rewrite the expression using the difference of squares identity (a^2 - b^2) = (a - b)(a + b):

a^4 - 16 = (a^2)^2 - 4^2 = (a^2 - 4)(a^2 + 4)

Now, we can apply the difference of squares identity again:

(a^2 - 4)(a^2 + 4) = (a - 2)(a + 2)(a^2 + 4)

So the expression can be factored as (a - 2)(a + 2)(a^2 + 4).