Find polynomial function of degree 3, given -1,2,-8
I don't mind answering the questions, but I don't like having to figure out what the question is, as well.
I assume those three numbers are zeroes of the polynomial. So, by the Factor Theorem, the factors are
y = (x+1)(x-2)(x+8)
To find a polynomial function of degree 3 given the roots -1, 2, and -8, you can use the fact that if a number x is a root of a polynomial, then (x - a) is a factor of the polynomial.
Let's denote the polynomial as f(x), and since we know the roots, we can write these factors as (x + 1), (x - 2), and (x + 8).
Now, we can write the polynomial function as:
f(x) = a(x + 1)(x - 2)(x + 8)
where a is a constant to be determined.
To find the value of a, let's substitute any of the given values of x into the polynomial. Let's use x = -1:
f(-1) = a((-1) + 1)((-1) - 2)((-1) + 8)
f(-1) = a(0)(-3)(7)
f(-1) = 0
Since f(-1) = 0, this means that (-1 + 1) is a factor of the polynomial.
Next, let's substitute x = 2:
f(2) = a((2) + 1)((2) - 2)((2) + 8)
f(2) = a(3)(0)(10)
f(2) = 0
Again, we have f(2) = 0, so (2 - 2) is a factor of the polynomial.
Lastly, let's substitute x = -8:
f(-8) = a((-8) + 1)((-8) - 2)((-8) + 8)
f(-8) = a(-7)(-10)(0)
f(-8) = 0
Similarly, f(-8) = 0, so (-8 + 8) is a factor of the polynomial.
Since we have three factors that are equal to zero, we know that the polynomial can be factored as:
f(x) = a(x + 1)(x - 2)(x + 8)
Now, we can simplify this polynomial and determine the value of a by expanding the equation:
f(x) = a(x + 1)(x - 2)(x + 8)
f(x) = a(x^2 - 2x + x - 2)(x + 8)
f(x) = a(x^2 - x - 2)(x + 8)
f(x) = a(x^3 + 8x^2 - x^2 - 8x - 2x - 16)
f(x) = a(x^3 + 7x^2 - 10x - 16)
Therefore, the polynomial function of degree 3, with roots -1, 2, and -8, is f(x) = a(x^3 + 7x^2 - 10x - 16), where a is any non-zero constant.
To find a polynomial function of degree 3, we need three points as input. In this case, the given points are (-1, 2, -8).
Let's assume the polynomial function is of the form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants to be determined.
Step 1: Substitute the x-values into the function:
f(-1) = a(-1)^3 + b(-1)^2 + c(-1) + d = -a + b - c + d = 2
f(2) = a(2)^3 + b(2)^2 + c(2) + d = 8a + 4b + 2c + d = 2
f(-8) = a(-8)^3 + b(-8)^2 + c(-8) + d = -512a + 64b - 8c + d = -8
Step 2: Solve the system of equations:
From the first equation, we have: -a + b - c + d = 2
From the second equation, we have: 8a + 4b + 2c + d = 2
From the third equation, we have: -512a + 64b - 8c + d = -8
By solving this system of equations, we will find the values of a, b, c, and d, which will give us the polynomial function of degree 3.