Factor Theorem & Rational Root Theorem

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Rational Root Theorem.

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Presentation transcript:

Factor Theorem & Rational Root Theorem Objective: SWBAT find zeros of a polynomial by using Rational Root Theorem

The Factor Theorem: For a polynomial P(x), x – a is a factor iff P(a) = 0 iff “if and only if” It means that a theorem and its converse are true

If P(x) = x3 – 5x2 + 2x + 8, determine whether x – 4 is a factor. remainder is 0, therefore yes other factor

Terminology: Solutions (or roots) of polynomial equations Zeros of polynomial functions “r is a zero of the function f if f(r) = 0” zeros of functions are the x values of the points where the graph of the function crosses the x-axis (x-intercepts where y = 0)

Ex 1: A polynomial function and one of its zeros are given, find the remaining zeros:

Ex 2: A polynomial function and one of its zeros are given, find the remaining zeros:

Rational Root Theorem: Suppose that a polynomial equation with integral coefficients has the root p/q , where p and q are relatively prime integers. Then p must be a factor of the constant term of the polynomial and q must be a factor of the coefficient of the highest degree term. (useful when solving higher degree polynomial equations)

Solve using the Rational Root Theorem: 4x2 + 3x – 1 = 0 (any rational root must have a numerator that is a factor of -1 and a denominator that is a factor of 4) factors of -1: ±1 factors of 4: ±1,2,4 possible rational roots: (now use synthetic division to find rational roots) (note: not all possible rational roots are zeros!)

Ex 3: Solve using the Rational Root Theorem: possible rational roots:

Ex 4: Solve using the Rational Root Theorem: possible rational roots:

Ex 5: Solve using the Rational Root Theorem: possible rational roots: To find other roots can use synthetic division using other possible roots on these coefficients. (or factor and solve the quadratic equation)