6.5 Theorems About Roots of Polynomials
Rational Root Theorem
If p/q is in simplest form and is a rational root of the polynomial equation anxn + an­1xn­1 + ... +a1x +a0 = 0 with integer
coefficients, then p must be a factor of a0 and
q must be a factor of an.
Translation...
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Examples: Use the rational roots theorem to list all possible rational roots and then find any actual roots.
a) 2x3 ­ 15x2 + 22x + 15 = 0
b) x3 + x2 ­ 3x ­ 3 = 0
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We can use the rational roots theorem to help us find
the zeros of a polynomial...
Example: Find the roots of 5x3 ­ 24x2 + 41x ­ 20 = 0.
Irrational Root Theorem
Let a and b be rational numbers and let √b be an irrational number. If a + √b is a root of a polynomial equation with rational coefficients, then the conjugate
a ­ √b also is a root.
Example: Finding Irrational Roots
a) A polynomial equation with rational
coefficients has the roots 3 ­ √5 and ­√7. Find two additional roots.
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Imaginary Root Theorem
If the imaginary number a + bi is a root of a polynomial equation with real coefficients, then the conjugate a ­ bi also is a root.
Example: Finding Imaginary Roots
a) A polynomial equation with real coefficients has
the roots of 8i and 3 + 5i. Find two additional roots.
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Example: Find a third degree polynomial equation with rational coefficients that has roots of ­2 and 3 ­ i.
6.5 p. 339 #1, 4, 7, 8, 10, 11, 13­18, 20, 21, 29, 30
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