Find the zeros of f(x) = x^3 - 5x^2 - x +5 by factoring
To find the zeros of the function f(x) = x^3 - 5x^2 - x + 5 by factoring, we need to find the values of x that make the function equal to zero.
We start by factoring out the common factor, which is 1:
f(x) = x^3 - 5x^2 - x + 5
= (x^3 - 5x^2) - (x - 5)
= x^2(x - 5) - (x - 5)
= (x - 5)(x^2 - 1)
Now we have a difference of squares, which can be factored further:
f(x) = (x - 5)(x - 1)(x + 1)
To find the zeros, we set each factor equal to zero and solve for x:
x - 5 = 0 -> x = 5
x - 1 = 0 -> x = 1
x + 1 = 0 -> x = -1
Therefore, the zeros of f(x) = x^3 - 5x^2 - x + 5 are x = 5, x = 1, and x = -1.
To find the zeros of the function f(x) = x^3 - 5x^2 - x + 5 by factoring, we can start by rearranging the terms and looking for common factors.
f(x) = x^3 - 5x^2 - x + 5
= (x^3 - 5x^2) + (-x + 5)
= x^2(x - 5) - 1(x - 5)
= (x^2 - 1)(x - 5)
Here we can see that (x^2 - 1) and (x - 5) are both factors of f(x).
Further factoring:
(x^2 - 1) = (x + 1)(x - 1)
So, the factored form of f(x) is:
f(x) = (x + 1)(x - 1)(x - 5)
To find the zeros of the function, we will set each factor equal to zero and solve for x:
x + 1 = 0
=> x = -1
x - 1 = 0
=> x = 1
x - 5 = 0
=> x = 5
Therefore, the zeros of the function f(x) = x^3 - 5x^2 - x + 5 are x = -1, x = 1, and x = 5.
To find the zeros of a polynomial function by factoring, we need to first set the function equal to zero (f(x) = 0). In this case, we have:
x^3 - 5x^2 - x + 5 = 0
Now, we can attempt to factor the expression. The first step is to look for any common factors. We can observe that 1 is a factor of each term in the polynomial:
x^3 - 5x^2 - x + 5 = (x^3 + x) + (-5x^2 + 5)
Next, factor out the common term from each pair of terms:
= x(x^2 + 1) - 5(x^2 - 1)
Now, let's focus on factoring the remaining quadratic expressions:
= x(x^2 + 1) - 5(x^2 - 1)
= x(x^2 + 1) - 5(x + 1)(x - 1)
Now, we have factored the polynomial into three terms: x, (x^2 + 1), and (x + 1)(x - 1). To find the zeros, we can set each term equal to zero and solve for x:
1) x = 0
2) x^2 + 1 = 0
This equation has no real solutions since the square of a real number is always positive. However, if we consider complex numbers, the solutions are x = ±i, where i is the imaginary unit.
3) (x + 1)(x - 1) = 0
Set each factor equal to zero:
x + 1 = 0 --> x = -1
x - 1 = 0 --> x = 1
Therefore, the zeros of the function f(x) = x^3 - 5x^2 - x + 5 are x = 0, x = -1, and x = 1.