Math

posted by .

Find a polynomial function with integer coefficients that has the given zeros.

0,0,4,1+i

Please explain!! I am very confused.

  • Math -

    For each root, you have the factor (x-a) where a is the root.

    So, you have (x-0)(x-0)(x-4)(x-1-i)

    But to get integral coefficients, you must also use the root 1-i to remove the imaginary numbers.

    So, now you have (x-0)(x-0)(x-4)(x-1-i)(x-1+i)

    = (x)(x)(x-4)(x-1-i)(x-1+i)

    Foil the complex numbers.

    = (x^2)(x-4)(x^2-x-ix-x+1+i+ix-i-i^2)
    = (x^2)(x-4)(x^2-2x+2)

    Now foil to get the final polynomial.

    = (x^2)(x^3-2x^2+2x-4x^2+8x-8)
    = (x^2)(x^3-6x^2+10x-8)
    = x^5 - 6x^4 + 10x^3 - 8x^2

  • Math -

    Find all the zeros of the function and write the polynomial as a product of linear factors.

    g(x)=x^5 - 8x^4 + 28x^3 - 56x^2 + 64x - 32

    The zero I found so far is 2. I am confused on how to find the rest.

  • Math -

    You can easily see that x = 2 is the only rational root witjout trying out all the root candidates. The trick is to choose a value for x for which g(x) is number with few factors. E.g.,:

    g(1) = -3

    This means that the polynomial

    P(x) = g(1 + x)

    is of the form:

    P(x) = x^5 + a x^4 + b x^3 + c x^2 +
    d x - 3

    I.e. we know that the coefficient of x^5 is 1 and that the constant term is
    -3. The fact that the constant term is minus 3 follows from the fact that it must be equal to P(0) which is g(1).

    The fact that the coefficient of x^5 is 1 follws from the fact that if you replace x^n by (1+x)^n and expand, the highest power will ave a coefficient of 1, so the coefficient of the highest power of the polynomial will be unchanged by the substitution.

    If we then apply the Rational Roots Theorem to P(x), then we see that the seroes must be divisors of 3, so the possible roots are:

    x = 3, x= -3, x= 1 and x = -1

    Since P(x)= g(1+x), this means that the possible zeros of g(x) are obtained by adding 1 to these root candidates:

    x = 4, x = -2, x = 2, x = 0

    If we apply the Rational Roots theorem to g(x), we find that the roots must be powers of 2. So, the only possible rational roots are:

    x = 4, x = -2, x = 2.

    But only x = 2 works.

    Then divide g(x) by x - 2 to obtain the polynomial:

    x^4 - 6 x^3 + 16 x^2 - 24 x +16

    You can then see that x = 2 is also a root, dividing by x - 2 gives you a third degree polynomial which also has x = 2 as a root, dividing agains gives you a quadratic equation that you can easily solve.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Algebra

    Can someone please explain how to do these problems. 1)write a polynomial function of least degree with intregal coefficients whose zeros include 4 and 2i. 2)list all of the possible rational zeros of f(x)= 3x^3-2x^2+7x+6. 3)Find all …
  2. Polynomial Function

    Could you help me with the following problem, I don't understand how to do it. Find an nth degree polynomial function with real coefficients satisfying the given conditions. 1. n=3; 3 and i are zeros; f(2)=20
  3. Polynomial Function

    Could you please check my answers? Find an nth degree polynomial function with real coefficients satisfying the given conditions. 1. n=3; 3 and i are zeros; f(2)=20 -I got: f(x)=-4^3+12x^2-4x+12 3.n=3;4 and i zeros;f(-3)=60 -I got:f(x)=6x^3+24x^2+6x+24
  4. Algebra

    Could you please check my answers? Find an nth degree polynomial function with real coefficients satisfying the given conditions. 1. n=3; 3 and i are zeros; f(2)=20 -I got: f(x)=-4^3+12x^2-4x+12 3.n=3;4 and i zeros;f(-3)=60 -I got:f(x)=6x^3+24x^2+6x+24
  5. algebra

    Find an equation of a polynomial function of degree 5 with integer coefficients with zeros 0, -2, and 1/2.
  6. math

    Find an equation of a polynomial function of degree 5 with integer coefficients with zeros 0, -2, and 1/2.
  7. Math

    Use the given information about a polynomial whose coefficients are real numbers to find the remaining zeros of the polynomial. Degree: 6 Zeros: -6 + 13i^3, -8 + s^2i, -3 - 4i
  8. calculus--please help!!

    I have two questions that I don't understand and need help with. 1. information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zerosof f. degree 4, zeros i;9+i 2. form a polynomial f(x) with …
  9. Pre-Calculus

    Find a polynomial with integer coefficients that satisfies the given conditions. P has degree 2 and zeros 2 + i and 2 − i.
  10. math (precalc)

    Find a polynomial with integer coefficients that satisfies the given conditions. R has degree 4 and zeros 3 − 4i and 5, with 5 a zero of multiplicity 2.

More Similar Questions