If (x-1),(x+1) and (x-2) are factors of the polynomial ax^3+bx^2+cx-1.
Find a,b and c respectively?
Let f(x) = ax^3 + bx^2 + cx - 1.
If (x - p) is a factor of f(x), then f(p) = 0.
Since (x-1) is a factor of f(x), then f(1) = 0
f(1) = a(1)^3 + b(1)^2 + c(1) - 1 = 0
a + b + c - 1 = 0..........Equation 1
Repeat this process for (x + 1) and (x - 2), respectively, to get:
-a + b - c - 1 = 0..........Equation 2
8a + 4b + 2c - 1 = 0.........Equation 3
Solve this system of 3 equations in 3 unknowns to find a, b, and c.
Well, let's start by using the properties of polynomial factors. Since (x-1), (x+1), and (x-2) are factors of the polynomial, it means that when we substitute x = 1, x = -1, and x = 2, the polynomial will equal zero.
So, let's substitute x = 1 into the polynomial:
a(1)^3 + b(1)^2 + c(1) - 1 = 0
This simplifies to:
a + b + c - 1 = 0
Now, let's substitute x = -1:
a(-1)^3 + b(-1)^2 + c(-1) - 1 = 0
-a + b - c - 1 = 0
Finally, let's substitute x = 2:
a(2)^3 + b(2)^2 + c(2) - 1 = 0
8a + 4b + 2c - 1 = 0
Now we have a system of three equations:
a + b + c = 1
-a + b - c = 1
8a + 4b + 2c = 1
And I'm sorry to disappoint you, but there is no funny answer to this question. Solving this system of equations will give us the values of a, b, and c, which are not obvious from the information given.
To find the values of a, b, and c, we will use the fact that if (x-1), (x+1), and (x-2) are factors of the polynomial ax^3+bx^2+cx-1, then the product of these factors equals the given polynomial.
First, let's expand the polynomial using the given factors:
(x-1)(x+1)(x-2) = ax^3+bx^2+cx-1
Expanding the left side of the equation:
(x-1)(x+1)(x-2) = (x^2-1)(x-2)
= x(x^2-1)-2(x^2-1)
= x^3-x-2x^2+2
= x^3-2x^2-x+2
Now, we can equate the expanded form of the polynomial to the given polynomial:
ax^3+bx^2+cx-1 = x^3-2x^2-x+2
Comparing the coefficients of like terms on both sides, we can set up the following equations:
For the x^3 term: a = 1
For the x^2 term: b = -2
For the x term: c = -1
For the constant term: -1 = 2 (which is not possible)
Since the constant terms do not match, there must be an error in the question or the given polynomial. Please check the problem and provide the correct information if possible.
To find the values of a, b, and c, we can use the fact that if (x-1), (x+1), and (x-2) are factors of the polynomial, then when we substitute these values into the polynomial, the result will be equal to zero.
Substituting x = 1 into the polynomial:
a(1)^3 + b(1)^2 + c(1) - 1 = 0
a + b + c - 1 = 0 --(1)
Substituting x = -1 into the polynomial:
a(-1)^3 + b(-1)^2 + c(-1) - 1 = 0
-a + b - c - 1 = 0 --(2)
Substituting x = 2 into the polynomial:
a(2)^3 + b(2)^2 + c(2) - 1 = 0
8a + 4b + 2c - 1 = 0 --(3)
We now have a system of three linear equations (equations 1, 2, and 3) with three variables (a, b, c). We can solve this system to find the values of a, b, and c.
Adding equations (1) and (2) eliminates the b term:
2b - 2 = 0
2b = 2
b = 1
Substituting b = 1 into equation (1):
a + 1 + c - 1 = 0
a + c = 0
a = -c
Substituting b = 1 into equation (3):
8a + 4(1) + 2c - 1 = 0
8a + 2c + 3 = 0
8(-c) + 2c + 3 = 0
-8c + 2c + 3 = 0
-6c + 3 = 0
-6c = -3
c = 0.5
Since a = -c, then a = -0.5.
Therefore, the values of a, b, and c are a = -0.5, b = 1, and c = 0.5, respectively.