y=x^3-2x^2+25x-50 Can someone please help me? I sort of understand it but need the steps and answer. This is a similar question to the real one to have an example. Thanks!
You can either use the factor theorem or long divide to find the roots : )
I would suggest checking when x = 2
This will help you find the first factor : )
Ms Pi 3.1415926358979323Thank you. I don't exactly understand what you're saying though. Is there any way you can start the steps and I try to finish the rest?
You can do grouping which is basically grouping the first two and second two terms.
step 1: (x^3-2x^2)(+25x-50)
step 2: take out any common factors x^2 (x-2) +25 (x-2)
step 3: group the factors together (x^2 +25) (x-2)
step 4: factor everything out
You can group the two first terms and two last terms because the ratios of the coefficients are equal. So treat the problem as y=(x^3-2x^2)+(25x-50) and then factor out an x^2 from the first group and a 25 from the second group and then combine then with what’s left to get (x^2+25)(x-2) and then you get x=2, -5i, and 5i
Of course! I can help you with that.
To find the answer, we can start by factoring the given cubic equation.
Step 1: Look for common factors
In this case, we don't have any common factors among all the terms.
Step 2: Use the Rational Root Theorem (optional)
The Rational Root Theorem helps us find potential solutions (roots) of the equation. It states that if a polynomial equation has a rational root, it can be expressed as a fraction of two integers, p/q, where p is a factor of the constant term (in this case, -50), and q is a factor of the leading coefficient (in this case, 1).
By testing out the potential rational roots using synthetic division or by substituting the values, we can find that x = 2 and x = -5 are both roots of the given equation.
Step 3: Divide the equation by the roots found (synthetic division)
Divide the original equation by (x - 2) and (x + 5) using synthetic division or long division.
(x^3 - 2x^2 + 25x - 50) / (x - 2) = x^2 + x + 27
(x^2 + x + 27) / (x + 5) = x^2 - 4x + 32
After the division, we get (x^2 - 4x + 32) as the quotient.
Step 4: Solve the quadratic equation
To solve the resulting quadratic equation, we can use factoring, completing the square or the quadratic formula. However, in this case, the quadratic equation x^2 - 4x + 32 can't be factored easily and doesn't have real roots.
Therefore, the original cubic equation y = x^3 - 2x^2 + 25x - 50 does not have any real roots or x-intercepts.
To summarize, the given equation does not have any real roots.