x^4-16x^3+90x^2-111x+173
When I put this equation into the graphing calculator, I get a parabola. I'm not sure how to find the zeros. Can you help?
Actually, It is not a parabola, it is a W shape. I try to adjust my window and still can't see it on the screen ok. please help
You're right, it is a W-shape, and in this case, it is more like a flattened parabola. The curve never touched the x-axis, so there are no real roots. In other words, all four roots are complex.
Are you familiar with solving a quartic? and working with complex numbers?
No, I have never solved a quartic nor am I familiar with working with complex numbers. The only thing I have used is the rational roots theorem and that doesn't work here.
Does that mean it can't be solved on the graphing calculator? Is there any way for you to guide me in the right direction?
The zeroes of the polynomial function are complex, so the expression cannot be factorized with rational coefficients.
If you are interested in the complex roots, you can try the following link:
http://www.sosmath.com/algebra/factor/fac12/fac12.html
Otherwise, the graph tells you that there are no real roots.
If you want to try the quartic formula, first try it with an equation that gives real roots. Once you have mastered it, you can try it with the above equation.
Sure, I can help you with that! To find the zeros (or x-intercepts) of a polynomial function like the one you've provided, you can use the graphing calculator or manually solve the equation by factoring or using other techniques.
If you want to find the zeros using a graphing calculator, here's what you can do:
1. Enter the equation into your graphing calculator.
2. Graph the equation to see the parabola.
3. Locate the points where the parabola crosses the x-axis. These are the zeros of the equation.
However, if you want to find the zeros manually without a graphing calculator, you can follow these steps:
1. Start by factoring out common factors, if possible. In this case, there are no common factors to factor out.
2. Look for any rational zeros using the Rational Root Theorem. The Rational Root Theorem states that if a rational number p/q is a zero of the polynomial equation, then p must be a factor of the constant term (173 in your case), and q must be a factor of the leading coefficient (1, since the coefficient of x^4 is usually considered as 1).
You can try various combinations of factors of 173 (±1, ±173) divided by factors of 1 (±1) to check if any of them give zeros. For example, you can try x = 1, -1, 173, -173.
3. Use synthetic division or long division to divide the polynomial by the linear factors you find in step 2. Any remainder of 0 indicates that the linear factor is a zero.
For instance, if you find that x - 1 is a factor, perform synthetic division or long division to divide the polynomial by x - 1 and check if the remainder is 0. Repeat this process for all possible linear factors until you find the zeros.
4. Once you find the zeros, you can find the corresponding factors and expression of the polynomial.
Remember, finding the zeros of a polynomial may require using a combination of these methods and can sometimes be a lengthy process. But the graphing calculator can be a helpful tool to visualize the graph and locate the zeros quickly.