Question

4х^3 - 24х^2 -4х +120= 4(х+2)(х-3)(х+а) найти а

Answer 1

4x^3-24x^2-4x+120=4(x+2)(x-3)(x+a)
4(x^3-6x^2-x+30)=4(x+2)(x-3)(x+a)
(x^3-6x^2-x+30)=(x+2)(x-3)(x+a)
раскроем первые две скобки справа от знака равенства
(x+2)(x-3)=x^2-3x+2x-6=x^2-x-6
(x^3-6x^2-x+30)=(x^2-x-6)(x+a)
так как имеем равенство, то левая часть равенства имеют такие же два множителя-скобки
выделим слева такое же выражение, как и в первой скобке справа
(x^3-x^2-5x^2-6x+5x+30)=(x^2-x-6)(x+a)
здесь в левой части равенства -6x^2 расписали как -x^2-5x^2, а слагаемое -x как -6x+5x
((x^3-x^2-6x)-5x^2+5x+30)=(x^2-x-6)(x+a)
(x(x^2-x-6)-5(x^2-x-6))=(x^2-x-6)(x+a)
в левой части равенства как общий множитель выносим за скобку
(x^2-x-6)(x-5)=(x^2-x-6)(x+a) 
выражения в первых скобках слева и справа равны, следовательно равны и выражения во второй скобке слева и справа
x-5=x+a
a=-5

Answer 2

To find the value of "a" in the equation, we can first factor the given expression on the left side of the equation and then compare it with the factored expression on the right side.

Given equation: 4x^3 - 24x^2 - 4x + 120 = 4(x + 2)(x - 3)(x + a)

Let's start by factoring the left-hand side (LHS) of the equation:

4x^3 - 24x^2 - 4x + 120 = 4(x^3 - 6x^2 - x + 30)

Now, we need to equate the two factored expressions:

4(x^3 - 6x^2 - x + 30) = 4(x + 2)(x - 3)(x + a)

Now, compare the corresponding coefficients of each power of "x":

For the cubic term: Coefficient on the LHS = 1 (from x^3) Coefficient on the RHS = 1 (from (x + 2)(x - 3)(x + a))

For the quadratic term: Coefficient on the LHS = -6 (from -6x^2) Coefficient on the RHS = (Sum of the cross-products of all possible pairs from (x + 2)(x - 3)(x + a))

For the linear term: Coefficient on the LHS = -1 (from -x) Coefficient on the RHS = (Sum of the cross-products of all possible pairs from (x + 2)(x - 3)(x + a))

For the constant term: Constant term on the LHS = 30 Constant term on the RHS = (Constant term from (x + 2)) * (Constant term from (x - 3)) * (Constant term from (x + a))

Now, the coefficient on the RHS for the quadratic term is the sum of cross-products of the three terms in the trinomial, and the coefficient on the RHS for the linear term is the sum of cross-products of the two terms from the trinomial and the "a" term. Thus, we get:

-6 = (2 * -3 * a) + (2 * a * -3) + (a * 2 * -1) -6 = -6a - 6a - 2a -6 = -14a

Now, solve for "a":

a = -6 / -14 a = 3/7

So, the value of "a" is 3/7.