Ask Question
2 October, 05:55

If a polynomial has one root in the form a+√b, it has a second root in the form of a_√b

+1
Answers (1)
  1. 2 October, 09:39
    0
    When roots of polynomials occur in radical form, they occur as two conjugates.

    That is,

    The conjugate of (a + √b) is (a - √b) and vice versa.

    To show that the given conjugates come from a polynomial, we should create the polynomial from the given factors.

    The first factor is x - (a + √b).

    The second factor is x - (a - √b).

    The polynomial is

    f (x) = [x - (a + √b) ]*[x - (a - √b) ]

    = x² - x (a - √b) - x (a + √b) + (a + √b) (a - √b)

    = x² - 2ax + x√b - x√b + a² - b

    = x² - 2ax + a² - b

    This is a quadratic polynomial, as expected.

    If you solve the quadratic equation x² - 2ax + a² - b = 0 with the quadratic formula, it should yield the pair of conjugate radical roots.

    x = (1/2) [ 2a + / - √ (4a² - 4 (a² - b) ]

    = a + / - (1/2) * √ (4b)

    = a + / - √b

    x = a + √b, or x = a - √b, as expected.
Know the Answer?
Not Sure About the Answer?
Get an answer to your question ✅ “If a polynomial has one root in the form a+√b, it has a second root in the form of a_√b ...” in 📙 Mathematics if there is no answer or all answers are wrong, use a search bar and try to find the answer among similar questions.
Search for Other Answers