Rational Zeros
Key Questions
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Answer:
See explanation...
Explanation:
The rational zeros theorem can be stated:
Given a polynomial in a single variable with integer coefficients:
#a_n x^n + a_(n-1) x^(n-1) + ... + a_0# with
#a_n != 0# and#a_0 != 0# , any rational zeros of that polynomial are expressible in the form#p/q# for integers#p, q# with#p# a divisor of the constant term#a_0# and#q# a divisor of the coefficient#a_n# of the leading term.Interestingly, this also holds if we replace "integers" with the element of any integral domain. For example it works with Gaussian integers - that is numbers of the form
#a+bi# where#a, b in ZZ# and#i# is the imaginary unit.
George C.
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Aug 7 2018
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The Rational Zeros Theorem states: If
#P(x)# is a polynomial with integer coefficients and if#p/q# is a zero of#P(x)# , (# P(p/q) = 0 # ), then#p# is a factor of the constant term of#P(x)# and q is a factor of the leading coefficient of P(x) .
Konstantinos Michailidis
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1
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Sep 17 2015
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Answer:
See explanation...
Explanation:
A polynomial in a variable
#x# is a sum of finitely many terms, each of which takes the form#a_kx^k# for some constant#a_k# and non-negative integer#k# .So some examples of typical polynomials might be:
#x^2+3x-4# #3x^3-5/2x^2+7# A polynomial function is a function wholse values are defined by a polynomial. For example:
#f(x) = x^2+3x-4# #g(x) = 3x^3-5/2x^2+7# A zero of a polynomial
#f(x)# is a value of#x# such that#f(x) = 0# .For example,
#x=-4# is a zero of#f(x) = x^2+3x-4# .A rational zero is a zero that is also a rational number, that is, it is expressible in the form
#p/q# for some integers#p, q# with#q != 0# .For example:
#h(x) = 2x^2+x-1# has two rational zeros,
#x=1/2# and#x=-1# Note that any integer is a rational number since it can be expressed as a fraction with denominator
#1# .
George C.
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May 30 2016
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You can use the rational root theorem:
Given a polynomial of the form:
#a_0x^n+a_1x^(n-1)+...+a_n# with#a_0,...,a_n# integers,all rational roots of the form
#p/q# written in lowest terms (i.e. with#p# and#q# having no common factor) will satisfy.#p | a_n# and#q | a_0# That is
#p# is a divisor of the constant term and#q# is a divisor of the coefficient of the highest order term.This gives you a finite number of possible rational roots to try.
For example, the rational roots of
#6x^4-7x^3+x^2-7x-5=0# must be of the form
#p/q# where#p# is#+-1# or#+-5# and
#q# is#1# ,#2# ,#3# or#6# .You can try substituting each of the possible combinations of
#p# and#q# as#x=p/q# into the polynomial to see if they work.In fact the only rational roots it has are
#-1/2# and#5/3# .Once you have found one root, you can divide the polynomial by the corresponding factor to simplify the problem.
George C.
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May 30 2015
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To find the zeroes of a function,
#f(x)# , set#f(x)# to zero and solve.
For polynomials, you will have to factor.For example: Find the zeroes of the function
#f(x) = x^2+12x+32# First, because it's a polynomial, factor it
#f(x) = (x+8)(x+4)# Then, set it equal to zero
#0 = (x+8)(x+4)# Set each factor equal to zero and the answer is
#x=-8# and#x=-4# .*Note that if the quadratic cannot be factored using the two numbers that add to this and multiple to be this method, then use the quadratic formula
#(-b +- sqrt(b^2-4ac))/(2a)# to factor an equation in the form of
#ax^2+bx+c# .Rahul Reddy · 2 · Jan 27 2015