Transcript Remainder Theorem • Let f(x) be an nth degree polynomial.

Remainder Theorem
• Let f(x) be an nth degree polynomial.
• If f(x) is divided by x – k, then the
remainder is equal to f(k).
• We can find f(k) using Synthetic Division.
Factor Theorem
• If f(k)= 0, then x – k is a factor of f(x).
• If x – k is a factor of f(x), then f(k) = 0
• Reminder: f(k) is the remainder of f(x)
divided by x – k.
Properties of Polynomials
• An nth degree polynomial has n linear
factors.
Ex) f(x)= x4 – 8x³+ 14x²+ 8x -15 =
( x -1)(x+1)(x -3)(x-5)
• An nth degree polynomial has n zeros. The
zeros could be complex.
Ex) f(x) = 2x³ - 4x² + 2x
3 zeros
100 zeros
Ex) f(x) = 3x100 + 2x85
Conjugate Pairs Theorem
• Let f(x) be an nth degree polynomial with
real coefficients.
• If a+bi is a zero of f(x), then the conjugate
a – bi must also be a zero of f(x).
• Ex) Let f(x) = x² - 4x +5
If f(2 + i) = 0, then f(2 – i) = 0
• Ex) Let f(x) = x³ + 2x² +x +2
f(i) = 0, f(-i) = 0, f( -2) = 0
Descartes Rule of Signs
• Let f(x) be a polynomial of the form
f(x) = anxn+an-1xn-1+…..a1x+a0
1) The number of positive real zeros of f(x) is
equal to the number of sign changes of f(x) or
is less than that number by an even integer.
2) The number of negative real zeros of f(x) is
equal to the number of sign changes in f(-x) or
is less than that number by an even integer.
Example
• Find all possible positive, negative real
and nonreal zeros of
f(x) = 4x4- 3x³ +5x² + x – 5
Rational Zero Theorem
• Let f(x) = anxn+an-1xn-1+…..a1x+a0
• If f(x) has rational zeros, they will be of the
form p/q, where
• p is a factor of a0 , and
• q is a factor of an
Example
• Find the list of all possible rational zeros for
each function below.
• A) f(x) = x³ + 3x² - 8x + 16
• B) f(x) = 3x4 + 14x³ - 6x² +x -12
• C) f(x) = 2x³ - 3x² + x – 6
Factoring for the finding Zeros of
Polynomials
• For 2nd degree, we factored or used the
quadratic formula.
x² - 3x – 10 = 0 , ( x – 5)(x + 2) = 0 so
x = 5 or x = -2.
For 3rd degree, we factored.
x³ - x² - 4x + 4 = 0 , x²(x -1) -4(x – 1) =0
( x – 1)(x² - 4) = 0 , (x – 1)(x - 2)(x + 2) =0
x = 1, x = 2, x = -2
But, Factoring by traditional means doesn’t always
work for all polynomials.
Strategy for Finding all the zeros of
a Polynomial
• Step 1: Use Descartes Rule of Signs
• Step 2: Use Rational Zeros Theorem to
get list of possible rational zeros.
• Step 3: From the list above, test which
ones make f(x) = 0.
• Do this using SYNTHETIC DIVISION!!!!
• Do not plug in the values into f(x)!!!
• We want to factor f(x) until we get a
quadratic function. Check Mate!!