Find all real zeros of the function

If the sum of the coefficients of a polynomial is zero then#1#is a zero. If the sum of the coefficients with signs inverted on the terms of odd degree is zero then#-1#is a zero.

Any polynomial with rational roots

Any rational zeros of a polynomial with integer coefficients of the form#a_n x^n + a_(n-1) x^(n-1) +...+ a_0#are expressible in the form#p/q#where#p, q#are integers,#p#a divisor of#a_0#and#q#a divisor of#a_n#.

Polynomials with degree <= 4

#ax+b = 0 => x = -b/a#

#ax^2+bx+c = 0 => x = (-b+-sqrt(b^2-4ac))/(2a)#

There are formulas for the general solution to a cubic, but depending on what form you want the solution in and whether the cubic has#1#or#3#Real roots, you may find some methods preferable to others.

In the case of one Real root and two Complex ones, my preferred method is Cardano's method. The symmetry of this method gives neater result formulations than Vieta's substitution.

In the case of three Real roots, it may be preferable to use the trigonometric substitution that squeezes a cubic into the identity#cos 3 theta = 4 cos^3 theta - 3 cos theta#, thereby finding zeros in terms of#cos#and#arccos#.

There are general formulas for the solution of quartic equations, but it's generally easier to work with the individual cases.

In the worst cases, you can transform#ax^4+bx^3+cx^2+dx+e#into a monic quartic by dividing by#a#, get into the form#t^4+pt^2+qt+r#using the substitution#t = x+b/(4a)#, then look at factorisations of the form:

#t^4+pt^2+qt+r = (t^2+At+B)(t^2-At+C)#

multiplying out and equating coefficients to get 3 simultaneous equations in#A#,#B#and#C#. Then use#(B+C)^2 = (C-B)^2+4BC#to derive a cubic equation in#A^2#. By now you hopefully know how to solve cubics, so you can find#A#, hence#B#and#C#, etc.

Answer link

George C.

Feb 20, 2016

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(x)#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(a_i)/(f'(a_i))#

For example, if#f(x) = x^5+x+3#, then#f'(x) = 5x^4+1#and you would iterate using the formula:

#a_(i+1) = a_i - (a_i^5+a_i+3)/(5a_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(x)#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.

One of the task in precalculus is finding zeros of the function - i.e. the intersection points with abscissa axis. Consider the graph of some function :

The zeros of the function are the points at which, as mentioned above, the graph of the function intersects the abscissa axis. To find the zeros of the function it is necessary and sufficient to solve the equation:

The zeros of the function will be the roots of this equation. Thus, the zeros of the function are at the point .

Our online calculator, based on Wolfram Alpha system is able to find zeros of almost any, even very complicated function.

A real zero of a function is a real number that makes the value of the function equal to zero.

A real number, r , is a zero of a function f , if f(r)=0 .

Example:

f(x)=x2−3x+2

Find x such that f(x)=0 .

0=x2−3x+2

0=(x−2)(x−1)

x=2  or  x=1

f(2)=22−3(2)+2=0

f(1)=12−3(1)+2=0

Since f(2)=0 and f(1)=0 , both 2 and 1 are real zeros of the function.

The zero of a function is any replacement for the variable that will produce an answer of zero. Graphically, the real zero of a function is where the graph of the function crosses the x‐axis; that is, the real zero of a function is the x‐intercept(s) of the graph of the function.

Example 1

Find the zeros of the function f ( x) = x 2 – 8 x – 9. 

Find x so that f ( x) = x 2 – 8 x – 9 = 0. f ( x) can be factored, so begin there. 

Therefore, the zeros of the function f ( x) = x 2 – 8 x – 9 are –1 and 9. This means 

f (–1) = 0 and f (9) = 0 

If a polynomial function with integer coefficients has real zeros, then they are either rational or irrational values. Rational zeros can be found by using the rational zero theorem.

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