given that the expression ax^3-11x^2+bx+2 is exactly divisible by 2x^2-3x-2, find a and b then factorise completly (do not use long division).
(cx-d)(2x^2 - 3x - 2) = ax^3 - 11x^2 + bx + 2
2cx^3 - (3c+2d)x^2 - (2c-3d)x + 2d = ax^3 - 11x^2 + bx + 2
2c = a
3c+2d = 11
2c-3d = -b
2d = 2
so,
d=1
3c+2=11
c=3
-3 = b
a = 6
check:
(3x-1)(2x^2 - 3x - 2) =
6x^3 - 11x^2 - 3x + 2
To find the values of a and b, we need to use the concept of polynomial division, also known as long division. However, you specified not to use long division. In that case, we need to approach the problem differently.
First, let's factorize the quadratic polynomial 2x^2 - 3x - 2. We can do so by finding its roots using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our quadratic polynomial 2x^2 - 3x - 2, the coefficients are:
a = 2, b = -3, c = -2
Using the quadratic formula, we can now calculate the roots:
x = (-(-3) ± √((-3)^2 - 4(2)(-2))) / (2(2))
= (3 ± √(9 + 16)) / 4
= (3 ± √25) / 4
Simplifying further:
x1 = (3 + 5) / 4 = 2
x2 = (3 - 5) / 4 = -1/2
Now that we have factored the quadratic polynomial, we use these roots to find the values of a and b in the cubic polynomial ax^3 - 11x^2 + bx + 2.
For x = 2:
0 = a(2)^3 - 11(2)^2 + b(2) + 2
0 = 8a - 44 + 2b + 2
0 = 8a + 2b - 42 (Equation 1)
For x = -1/2:
0 = a(-1/2)^3 - 11(-1/2)^2 + b(-1/2) + 2
0 = -1/8a - 11/4 + b/2 + 2
0 = -1/8a + b/2 - 33/4 (Equation 2)
We now have a system of equations (Equations 1 and 2) with two unknowns, a and b. We can solve this system of equations to find the values of a and b.
Multiplying Equation 1 by 4:
0 = 32a + 8b - 168 (Equation 3)
Multiplying Equation 2 by 8:
0 = -1a + 4b - 99 (Equation 4)
Now, we have two equations with just two unknowns, which we can solve. Adding Equation 3 and Equation 4:
0 = 32a + 8b - 168 + -1a + 4b - 99
0 = 31a + 12b - 267
Simplifying further:
31a + 12b = 267 (Equation 5)
Using this equation, we can solve for a and b. However, without further information or additional equations, we cannot determine the specific values of a and b to factorize the cubic polynomial completely.
Therefore, additional information is required to provide the factorized form of the given polynomial without using long division.