# Zeros

## Key Questions

• The zeros of a function represent the x value(s) that result in the y value being 0.

The zeros of a function represent the x-intercept(s) when the function is graphed.

The zeros of a function represent the root(s) of a function.

The zeros of a function represent the solution(s) of a function.

Graph the function on a graphing calculator to see what the x-coordinates are where the function intersects the x-axis.

#### Explanation:

The zeros of a function are found by determining what x-values will cause the y-value to be equal to zero. One way to find the zeros is to graph the function on a graphing calculator to see what the x-coordinates are where the function intersects the x-axis.

• A zero of a function is an interception between the function itself and the X-axis.
The possibilities are:

• no zero (e.g. $y = {x}^{2} + 1$) graph{x^2 +1 [-10, 10, -5, 5]}
• one zero (e.g. $y = x$) graph{x [-10, 10, -5, 5]}
• two or more zeros (e.g. $y = {x}^{2} - 1$) graph{x^2-1 [-10, 10, -5, 5]}
• infinite zeros (e.g. $y = \sin x$) graph{sinx [-10, 10, -5, 5]}

To find the eventual zeros of a function it is necessary to solve the equation system between the equation of the function and the equation of the X-axis ($y = 0$).

If algebraic solutions are not usable, try Newton's method or similar to find numeric approximations.

#### Explanation:

Quintics and other more complicated functions

If $f \left(x\right)$ is a well behaved continuous, differentiable function - e.g. a polynomial, then you can find its zeros using Newton's method.

Starting with an approximation ${a}_{0}$, iterate using the formula:

${a}_{i + 1} = {a}_{i} - f \frac{{a}_{i}}{f ' \left({a}_{i}\right)}$

For example, if $f \left(x\right) = {x}^{5} + x + 3$, then $f ' \left(x\right) = 5 {x}^{4} + 1$ and you would iterate using the formula:

${a}_{i + 1} = {a}_{i} - \frac{{a}_{i}^{5} + {a}_{i} + 3}{5 {a}_{i}^{4} + 1}$

Putting this into a spreadsheet with ${a}_{0} = - 1$, I got the values:

${a}_{0} = - 1$

${a}_{1} = - 1.166666666666667$

${a}_{2} = - 1.134701651504899$

${a}_{3} = - 1.133002126375077$

${a}_{4} = - 1.132997565917805$

${a}_{5} = - 1.132997565885065$

${a}_{6} = - 1.132997565885065$

If $f \left(x\right)$ has several Real zeros, then you may find them by choosing different values of ${a}_{0}$.

Newton's method can also be used to find Complex zeros in a similar way, but you may prefer to use methods like Durand-Kerner to find all zeros at once.

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