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Pre-Cal 30S January 14, 2009

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Applications of the remainder theorem and the Rational roots theorem.

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Pre-Cal 30S January 14, 2009

1. 1. Rational Roots Theorem (really this time) At the Feet of an Ancient Master by ﬂickr user premasagar
2. 2. Determine each value of k. (a) When x 3 + kx2 + 2x - 3 is divided by x + 2, the remainder is 1.
3. 3. Determine each value of k. (a) When x 3 + kx2 + 2x - 3 is divided by x + 2, the remainder is 1.
4. 4. Determine each value of k. (b) When x4 - kx 3 + 2x2 + x + 4 is divided by x - 3, the remainder is 16.
5. 5. When the polynomial 2x 2 + bx - 5 is divided by x - 3, the remainder is 7. (a) Determine the value of b. (b) What is the remainder when the polynomial is divided by x - 2?
6. 6. Rational Roots Theorem For any polynomial function if P(x) has rational roots, they may be found using this procedure: Example Procedure Step 1: Find all possible ƒ(x) = 3x3 - 4x2 - 5x + 2 numerators by listing the positive and negative 1, -1, 2, -2 factors of the constant term.
7. 7. Rational Roots Theorem For any polynomial function if P(x) has rational roots, they may be found using this procedure: Example Procedure ƒ(x) = 3x3 - 4x2 - 5x + 2 Step 2: Find all possible denominators by listing the positive factors of the 1, 3 leading coefﬁcient.
8. 8. Rational Roots Theorem For any polynomial function if P(x) has rational roots, they may be found using this procedure: Example Procedure ƒ(x) = 3x3 - 4x2 - 5x + 2 Step 3: List all possible rational roots. Eliminate 1, -1, 2, -2 all duplicates. 1, 3
9. 9. Rational Roots Theorem For any polynomial function if P(x) has rational roots, they may be found using this procedure: Example Procedure ƒ(x) = 3x3 - 4x2 - 5x + 2 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.) is a root! -1 So,
10. 10. Rational Roots Theorem For any polynomial function if P(x) has rational roots, they may be found using this procedure: Example Procedure Step 5: Factor the quadratic. Step 6: Find all roots.
11. 11. Rational Roots Theorem You try ... ƒ(x) = x3 + 3x 2 - 13x - 15