pre-cal 30s january 14, 2009
DESCRIPTION
Applications of the remainder theorem and the Rational roots theorem.TRANSCRIPT
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Rational Roots Theorem(really this time)
At the Feet of an Ancient Master by flickr user premasagar
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Determine each value of k.(a) When x + kx + 2x - 3 is divided by x + 2, the remainder is 1.3 2
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Determine each value of k.(a) When x + kx + 2x - 3 is divided by x + 2, the remainder is 1.3 2
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Determine each value of k.(b) When x - kx + 2x + x + 4 is divided by x - 3, the remainder is 16.4 3 2
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(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
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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
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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
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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
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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
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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.
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Rational Roots TheoremYou try ...
ƒ(x) = x + 3x - 13x - 153 2
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