Rational Zeros

Key Questions

  • What is the rational zeros theorem?

    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. · 1 · Aug 7 2018
  • What is the rational zeros test?

    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 · 1 · Sep 17 2015
  • What are the rational zeros of a polynomial function?

    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. · 1 · May 30 2016
  • How do you find all the rational zeros of a polynomial function?

    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. · 1 · May 30 2015
  • How do I find all the rational zeros of function?

    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

Questions

Real Zeros of Polynomials