1. 7/16/20159-5 9-5 The Factor Theorem

2. 7/16/20159-5 Factor Theorem Factor Theorem: For a polynomial f(x) a number c is a solution to f(x) = 0 iff (x – c) is a factor of f(x).

3. 7/16/20159-5 Putting it all together… Equivalent Statements: For any polynomial f, the following are logically equivalent statements: (x – c) is a factor of f(x).(x – c) is a factor of f(x). f(c) = 0f(c) = 0 c is an x-intercept of the graph of f(x)c is an x-intercept of the graph of f(x) c is a zero of f(x)c is a zero of f(x) The remainder when f(x) is divided by (x-c) is 0The remainder when f(x) is divided by (x-c) is 0 c is a solution to the equationc is a solution to the equation

4. Application: (Playing “tennis”) Factor by first finding one zero by graphing, and then dividing to find the others. Let’s begin…. 7/16/20159-5

5. Summary… Steps to the process: 1)Find a zero 2)Use the zero to get a factor 3)Use the factor to divide and make the polynomial smaller in degree 4)Use the quotient and factor it 5)Use those factors to find the remaining zeros 7/16/20159-5

6. 7/16/20159-5 Finding Polynomials To Find a polynomial when the zeros are known: Write a set of factors as (x – “zero”)Write a set of factors as (x – “zero”) Multiply the factors togetherMultiply the factors together Example: Zeros at x = -1, 2, 4 f(x) = (x + 1)(x – 2)(x – 4) f(x) = (x 2 – x – 2)(x – 4) f(x) = x 3 – 4x 2 – x 2 + 4x – 2x + 8 f(x) = x 3 – 5x 2 + 2x + 8