find all zeros of the function.
f(x) = x^3-10x^2+33x-34
I found 2, using my calculator. What do i do after that?
One you have one zero, say x = 5.3 (invented example number)
Then divide the original polynomial by (x-5.3)
That leaves you with a quadratic, which you can solve with your calculator.
How do i solve the quadratic with my calculator. Thanks so much for your help:)
Use the quadratic equation
if
y = a x^2 + b x + c
then y is zero if
x = -b/2a +or- (1/2a)sqrt(b^2 - 4 a c)
To find all the zeros of the function f(x) = x^3 - 10x^2 + 33x - 34, you can apply the Rational Root Theorem and synthetic division. Here's a step-by-step guide:
Step 1: Apply the Rational Root Theorem.
The Rational Root Theorem states that any rational root (zero) of a polynomial must be a divisor of the constant term (34) divided by a divisor of the leading coefficient (1). In this case, possible rational roots are ±1, ±2, ±17, or ±34.
Step 2: Use synthetic division.
Start by selecting one of the rational numbers you obtained in Step 1 and perform synthetic division with it. For example, let's start with x = 2.
Perform synthetic division as follows:
2 | 1 -10 33 -34
| 2 -16 34
|_________________
1 -8 17 0
The remainder is 0, which means 2 is a zero of the function f(x).
Step 3: Factor the quotient obtained in Step 2.
The quotient obtained from synthetic division is 1x^2 - 8x + 17. You can use factoring or the quadratic formula to factor it further.
However, in this case, the quadratic equation doesn't factor, so you can use the quadratic formula to find the remaining zeros.
Step 4: Use the quadratic formula.
The quadratic formula is given by x = (-b ± √(b^2 - 4ac)) / (2a).
For the equation 1x^2 - 8x + 17, substitute a = 1, b = -8, c = 17 into the quadratic formula, and solve for x.
x = (-(-8) ± √((-8)^2 - 4(1)(17))) / (2(1))
x = (8 ± √(64 - 68)) / 2
x = (8 ± √(-4)) / 2
Since the discriminant (√(-4)) is negative, there are no real solutions.
Therefore, the zeros of the function f(x) = x^3 - 10x^2 + 33x - 34 are x = 2 and the complex solutions obtained from the quadratic formula.