Find a fourth degree polynomial with given zeros calculator

 Quartic Polynomials Division Calculator

An 4th degree polynominals divide calcalution.

Fourth Degree Equation

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Detailed Answer  

Fourth Degree Polynomial Equations Formula

y = ax4 + bx3 + cx2 + dx + e

4th degree polynomials are also known as quartic polynomials.

Quartics has the following characteristics

1. Zero to 4 roots.

2. 1, 2 or 3 extrema.

3. Zero, one or two inflection points.

4. No general symmetry.

@Alex : There is no need for a calculator to draw 9 points on a graph.

You wrote : I know that the format of the equation should be $P(x)=ax^4+bx^3+cx^2+dx+e$.

Well, from what do you know this ? Is it in the wording of your exercise ? You should have given more information. Without one cannot show you the better way to solve your problem.

If it is asked for finding a good but approximate fit of the fourth degree polynomial curve to the points, the regression method is recommended. It can be computed without calculator, but this would be very tiresome. I will not discuss more about the methods to apply because several answers were already given on this subject.

If it is asked for finding a fourth polynomial curve passing exactly on the 9 points, it is impossible. It is obvious just looking at the graph. Nevertheless, a proof is shown below :

We see that four points have the same value $y=-3$. Changing of function $Y(x)=y(x)+3$ shows that the four points are at the four roots of the function $Y(x)=c(x+10)(x+5)(x-1)(x-5.5)$.

So, the four points are exactly on the curve
$$y(x)=-3+c(x+10)(x+5)(x-1)(x-5.5)$$
To make a fifth point $(x_5\:,\:y_5)$ exactly on the curve : $$c=\frac{y_5+3}{(x_5+10)(x_5+5)(x_5-1)(x_5-5.5)}$$ The equation of the fourth degree polynomial is : $$y(x)=-3+(y_5+3)\frac{(x+10)(x+5)(x-1)(x-5.5)}{(x_5+10)(x_5+5)(x_5-1)(x_5-5.5)} $$ The figure below shows the five cases : On each one, they are five points exactly on the curve and of course four remaining points far from the curve.

Quartic equation

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How many zeros can a 4th degree polynomial have?