# How do i find the roots of this?

• DeanBH
In summary, you need to factor the expression into linear binomial or a quadratic and a linear binomial.
DeanBH
y = X^3 - 4x^2 -7x + 10

i have to draw a graph of this, stationary points don't matter but all else does.

i know what it would look like because it is a cubic it will have 3 roots.

but i don't know how to find them, i forgot :P

You need to factor the expression into linear binomials or a quadratic and a linear binomial. Study synthetic division and rational roots theorem.

For this equation it's not that hard.The first thing I usually do is try to find roots manually. To find roots you need to solve y = 0.
$$x^3-4x^2-7x+10=0$$

Just try a few integers ranging from -3 to 3 or something and you will probably find one or two values for x that yield 0.

Let's say you found 1 of these values, x = a.
You now know that (x - a) is a factor in the factorized form of the equation:
$$(x-a)(...) = x^3-4x^2-7x+10$$

To find the remaining (...) you can use Long Division:
$$(...) = \frac{x^3-4x^2-7x+10}{x-a}$$An easier way, which afaik only works if you found two roots manually, let's say x = b and x = c, you can do this:
$$(x-b)(x-c)(x-y) = x^3-4x^2-7x+10$$ for some unknown y.

Factor out the brackets and you can find the value of y easily.

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There is no easy way to factor any polynomial- just look at the factors of constant term and then "trial and error". And cubics are much harder than quadratics.

That's why the first thing I would try is to "look for" simple zeros.

Very, very large hint: what is y(1)?

I ended up trial and erroring it to find that its -2 1 and 5, but yeh..

i thought there might have been a method. thx

Another hint, this equation allows you to find two roots easily so you don't need to use Long Division if you don't know how that works with polynomials.

EDIT
Too late...

What do you mean by a method?
Perhaps you mean some formula like the quadratic formula $$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$?
If so, these formulas do exist for third (cubic) and fourth (quartic) degree polynomials, but they are really hard and you won't even try to remember them.
Check out Wikipedia for derivations: http://en.wikipedia.org/wiki/Cubic_function

If you encounter the need to find the roots of a cubic in school or on an exam, the roots are usually integers or very easy numbers.

In real world problems however this is usually not the case and the method I described above does not work. Instead you would probably use a computer who does it in seconds...

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Try division using binomials of (x-1), (x-2), (x-5), (x+1), (x+2), (x+5), (x-10), (x+10). Use synthetic division if you know it because it is faster. When you divide and obtain a remainder of zero, you have found a linear binomial factor.

## 1. How do I find the roots of a polynomial equation?

To find the roots of a polynomial equation, you can use a few different methods such as factoring, graphing, or using the quadratic formula. The method you choose will depend on the complexity of the equation and your familiarity with each method.

## 2. Can I use a calculator to find the roots of an equation?

Yes, many scientific and graphing calculators have a built-in function for finding the roots of an equation. However, it's important to understand the principles behind finding roots in order to use the calculator effectively and to verify the accuracy of the results.

## 3. What are complex roots and how do I find them?

Complex roots are solutions to an equation that involve imaginary numbers. These roots are typically found in quadratic equations with a negative discriminant. To find complex roots, you can use the quadratic formula and substitute in the appropriate values for the coefficients.

## 4. How do I know if my answer is a root of the equation?

If you have found a potential root of an equation, you can check if it is a true root by plugging it back into the equation and seeing if it satisfies the equation. If the result is 0, then the number is a root of the equation.

## 5. Can I find the roots of any type of equation?

No, not all equations have real solutions or can be solved using traditional methods. Some equations may have complex roots or require advanced techniques such as Newton's method or numerical methods to approximate the roots. It's important to consider the type of equation and the methods available before attempting to find its roots.

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