pre-cal 30s january 14, 2009
DESCRIPTION
Applications of the remainder theorem and the Rational roots theorem.TRANSCRIPT
Rational Roots Theorem(really this time)
At the Feet of an Ancient Master by flickr user premasagar
Determine each value of k.(a) When x + kx + 2x - 3 is divided by x + 2, the remainder is 1.3 2
Determine each value of k.(a) When x + kx + 2x - 3 is divided by x + 2, the remainder is 1.3 2
Determine each value of k.(b) When x - kx + 2x + x + 4 is divided by x - 3, the remainder is 16.4 3 2
(b) What is the remainder when the polynomial is divided by x - 2?
(a) Determine the value of b.
When the polynomial 2x + bx - 5 is divided by x - 3, the remainder is 7.2
Rational Roots Theorem
if P(x) has rational roots, they may be found using this procedure:
Procedure ExampleStep 1: Find all possible numerators by listing the positive and negative factors of the constant term.
For any polynomial function
ƒ(x) = 3x - 4x - 5x + 23 2
1, -1, 2, -2
Rational Roots Theorem
if P(x) has rational roots, they may be found using this procedure:
Procedure Example
For any polynomial function
ƒ(x) = 3x - 4x - 5x + 23 2Step 2: Find all possible denominators by listing the positive factors of the leading coefficient.
1, 3
Rational Roots Theorem
if P(x) has rational roots, they may be found using this procedure:
Procedure Example
For any polynomial function
ƒ(x) = 3x - 4x - 5x + 23 2Step 3: List all possible rational roots. Eliminate all duplicates. 1, -1, 2, -2
1, 3
Rational Roots Theorem
if P(x) has rational roots, they may be found using this procedure:
Procedure Example
For any polynomial function
Step 4: Use synthetic division and the factor theorem to reduce ƒ(x) to a quadratic. (In our example, we’ll only need one such root.)
So,
-1 is a root!
ƒ(x) = 3x - 4x - 5x + 23 2
Rational Roots Theorem
if P(x) has rational roots, they may be found using this procedure:
Procedure Example
For any polynomial function
Step 5: Factor the quadratic.
Step 6: Find all roots.
Rational Roots TheoremYou try ...
ƒ(x) = x + 3x - 13x - 153 2