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# Jan. 12 Binomial Factor Theorm

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### Jan. 12 Binomial Factor Theorm

1. 1. Factor and Rational Roots Theorems
2. 2. Factor Theorem A Polynomial P(x) -only has a factor (x - a) if the value of P(a) is 0 (no remainder) Example 3 a) Determine whether x + 2 is a factor of f(x) = x - 6x - 4 b) Determine the other factors of f(x)
3. 3. Rational Roots Theorem For any polynomial function if P(x) has rational roots, they may be found using this procedure: Procedure Example 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.
4. 4. Rational Roots Theorem For any polynomial function if P(x) has rational roots, they may be found using this procedure: Procedure Example ƒ(x) = 3x3 - 4x2 - 5x + 2 Step 2: Find all possible denominators by listing the positive factors of the 1, 3 leading coefﬁcient.
5. 5. Rational Roots Theorem For any polynomial function if P(x) has rational roots, they may be found using this procedure: Procedure Example Step 3: List all possible ƒ(x) = 3x3 - 4x2 - 5x + 2 rational roots. Eliminate all duplicates. 1, -1, 2, -2 1, 3
6. 6. Rational Roots Theorem For any polynomial function if P(x) has rational roots, they may be found using this procedure: Procedure Example 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, weʼll only need one such root.) - 1 is a root! So,
7. 7. Rational Roots Theorem For any polynomial function if P(x) has rational roots, they may be found using this procedure: Procedure Example Step 5: Factor the quadratic. Step 6: Find all roots.
8. 8. Step 1: Find all possible Step 4: Use synthetic division and numerators by listing the the factor theorem to reduce ƒ(x) positive and negative to a quadratic. (In our example, factors of the constant weʼll only need one such root.) term. Step 2: Find all possible Step 5: Factor the quadratic. denominators by listing the positive factors of the leading coefﬁcient. Step 6: Find all roots. Step 3: List all possible rational roots. Eliminate all duplicates. You try!!! Find all of the factors and roots of this polynomial ƒ(x) = x3 + 3x 2 - 13x - 15