Question

ДАЮ 25 БАЛЛОВ!!! Разложите на множители: A) 75m2 - 30mn + 3n2 - 2n + 10m Б) x2 + х - 6 B) (a + b)• (a - b)3 - (a - b) (a + b)3 Прошу расписать решение, а не просто кратко написать или записать ответ.

Answer 1

A) 75m²- 30mn + 3n² - 2n + 10m=

(75m²- 30mn + 3n² )- 2n + 10m=3*(25m²- 10mn + n²)+2(5m- n)=

3*(5m-n)²+2(5m- n)=(3*(5m-n)+2)*(5m- n)=(15m-3n+2)*(5m- n)

(х²-2ху+у²)=(х-у)²- собрал по формуле. затем вынес скобку за одинаковую скобку.

Б) x² + х - 6=0, по теореме, обратной теореме Виета х₁=-3; х₂=2

значит,  x² + х - 6=(х+3)*(х-2)- это разложение квадратного трехчлена на множители ах²+bx+c=a*(x-x₁)(x-x₂)

B) (a + b)* (a - b)³ - (a - b)* (a + b)³- выносим одинаковые скобки с наименьшим показателем. затем применяем формулы квадрата суммы и разности (а±b)²=a²±2ab+b²;

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

(a + b)* (a - b)*( а²-2ab + b²-a² -2ab- b²)=(a + b)* (a - b)*( -4ab)=

-4аb*(a + b)* (a - b)

Answer 2

Problem A: Factoring 75m^2 - 30mn + 3n^2 - 2n + 10m

To factor the expression 75m^2 - 30mn + 3n^2 - 2n + 10m, we can look for common factors and then use factoring techniques.

First, let's look for common factors among the terms. We can see that both 75m^2 and 10m have a common factor of 5m. Similarly, -30mn and -2n have a common factor of -2n. However, the term 3n^2 does not have any common factors with the other terms.

Now, let's factor out the common factors: 75m^2 - 30mn + 3n^2 - 2n + 10m = 5m(15m - 6n) + n(3n - 2) + 10m

So, the factored form of the expression 75m^2 - 30mn + 3n^2 - 2n + 10m is 5m(15m - 6n) + n(3n - 2) + 10m.

Problem B: Factoring x^2 + x - 6

To factor the expression x^2 + x - 6, we need to find two numbers whose product is -6 and whose sum is 1 (the coefficient of the x term).

Let's list all the possible pairs of numbers whose product is -6: 1 * -6 = -6 -1 * 6 = -6 2 * -3 = -6 -2 * 3 = -6

Among these pairs, the pair that has a sum of 1 is 2 and -3. So, we can rewrite the expression as: x^2 + x - 6 = (x + 2)(x - 3)

Therefore, the factored form of the expression x^2 + x - 6 is (x + 2)(x - 3).

Problem C: Factoring (a + b) • (a - b)^3 + (a - b) • (a + b)^3

To factor the expression (a + b) • (a - b)^3 + (a - b) • (a + b)^3, we can use the formula for the difference of cubes, which states that a^3 - b^3 = (a - b)(a^2 + ab + b^2).

Let's apply this formula to the two terms in the expression: (a + b) • (a - b)^3 + (a - b) • (a + b)^3 = (a - b)(a^2 - 2ab + b^2) + (a + b)(a^2 + 2ab + b^2)

Now, let's simplify each term: (a - b)(a^2 - 2ab + b^2) + (a + b)(a^2 + 2ab + b^2) = a^3 - 2a^2b + ab^2 - a^2b + 2ab^2 - b^3 + a^3 + 2a^2b + ab^2 + a^2b + 2ab^2 + b^3 = 2a^3 + 4ab^2

So, the factored form of the expression (a + b) • (a - b)^3 + (a - b) • (a + b)^3 is 2a^3 + 4ab^2.

I hope this helps! Let me know if you have any further questions.