Difference between revisions of "Finding Roots of an Equation"
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| − | + | ''See [[Polynomial Functions]] for more on factoring and finding roots'' | |
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| + | A '''zero''' (also sometimes called a '''root''') of a real function <math>f</math>, is a member <math>x</math> of the domain of <math>f</math> such that <math>f(x)</math> ''vanishes'' at <math>x</math>; that is, the function <math>f</math> attains the value of 0 at <math>x</math>, or equivalently, <math>x</math> is the solution to the equation <math>f(x) = 0</math>. A "zero" of a function is thus an input value that produces an output of 0. | ||
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| + | A '''root''' of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial <math>f</math> of degree two, defined by | ||
| + | :<math>f(x)=x^2-5x+6</math> | ||
| + | has the two roots <math>2</math> and <math>3</math>, since | ||
| + | :<math>f(2) = 2^2 - 5 \cdot 2 + 6 = 0\quad\textrm{and}\quad f(3) = 3^2 - 5 \cdot 3 + 6 = 0</math>. | ||
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| + | If the function maps real numbers to real numbers, then its zeros are the <math>x</math>-coordinates of the points where its graph meets the ''x''-axis. An alternative name for such a point <math>(x,0)</math> in this context is an <math>x</math>-intercept. | ||
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| + | ==Solution of an equation== | ||
| + | Every equation in the unknown <math>x</math> may be rewritten as | ||
| + | |||
| + | :<math>f(x)=0</math> | ||
| + | |||
| + | by regrouping all the terms in the left-hand side. It follows that the solutions of such an equation are exactly the zeros of the function <math>f</math>. In other words, a "zero of a function" is precisely a "solution of the equation obtained by equating the function to 0", and the study of zeros of functions is exactly the same as the study of solutions of equations. | ||
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| + | == Polynomial roots == | ||
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| + | Every real polynomial of odd degree has an odd number of real roots (counting multiplicities); likewise, a real polynomial of even degree must have an even number of real roots. Consequently, real odd polynomials must have at least one real root (because the smallest odd whole number is 1), whereas even polynomials may have none. This principle can be proven by reference to the intermediate value theorem: since polynomial functions are continuous, the function value must cross zero, in the process of changing from negative to positive or vice versa (which always happens for odd functions). | ||
==Resources== | ==Resources== | ||
| + | * [https://en.wikipedia.org/wiki/Zero_of_a_function Zero of a function], Wikipedia | ||
* [https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/find-zeros-of-a-polynomial-function/ Finding Zeros of Polynomials], Lumen Learning | * [https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/find-zeros-of-a-polynomial-function/ Finding Zeros of Polynomials], Lumen Learning | ||
* [https://www.freemathhelp.com/finding-roots/#:~:text=A%20root%20is%20a%20value,f%20(%20x%20)%20%3D%200%20. Finding Roots], Free Math Help | * [https://www.freemathhelp.com/finding-roots/#:~:text=A%20root%20is%20a%20value,f%20(%20x%20)%20%3D%200%20. Finding Roots], Free Math Help | ||
* [https://www.cliffsnotes.com/study-guides/algebra/algebra-ii/polynomial-functions/zeros-of-a-function Zeros of a Function], Cliff's Notes | * [https://www.cliffsnotes.com/study-guides/algebra/algebra-ii/polynomial-functions/zeros-of-a-function Zeros of a Function], Cliff's Notes | ||
Latest revision as of 13:26, 18 October 2021
See Polynomial Functions for more on factoring and finding roots
A zero (also sometimes called a root) of a real function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , is a member of the domain of such that vanishes at ; that is, the function attains the value of 0 at , or equivalently, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is the solution to the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = 0} . A "zero" of a function is thus an input value that produces an output of 0.
A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} of degree two, defined by
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=x^2-5x+6}
has the two roots Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3} , since
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(2) = 2^2 - 5 \cdot 2 + 6 = 0\quad\textrm{and}\quad f(3) = 3^2 - 5 \cdot 3 + 6 = 0} .
If the function maps real numbers to real numbers, then its zeros are the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -coordinates of the points where its graph meets the x-axis. An alternative name for such a point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,0)} in this context is an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -intercept.
Solution of an equation
Every equation in the unknown Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} may be rewritten as
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=0}
by regrouping all the terms in the left-hand side. It follows that the solutions of such an equation are exactly the zeros of the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} . In other words, a "zero of a function" is precisely a "solution of the equation obtained by equating the function to 0", and the study of zeros of functions is exactly the same as the study of solutions of equations.
Polynomial roots
Every real polynomial of odd degree has an odd number of real roots (counting multiplicities); likewise, a real polynomial of even degree must have an even number of real roots. Consequently, real odd polynomials must have at least one real root (because the smallest odd whole number is 1), whereas even polynomials may have none. This principle can be proven by reference to the intermediate value theorem: since polynomial functions are continuous, the function value must cross zero, in the process of changing from negative to positive or vice versa (which always happens for odd functions).
Resources
- Zero of a function, Wikipedia
- Finding Zeros of Polynomials, Lumen Learning
- Finding Roots, Free Math Help
- Zeros of a Function, Cliff's Notes
