MATH 1414 Sections 3.3 and 3.4
If a polynomial in the form ( x-k ) is a factor of another polynomial, P ( x ) then ( x-k ) divides P ( x ) with remainder 0.
k is also called a zero of the polynomial P ( x ).
Example 1. Determine whether the second polynomial is a factor of the first.
Example 2. Verify that k =5 is a zero of
Example 3. Factor given that k =5 is a zero.
Example 4. Factor given that k =1 is a zero.
If zeros repeat we state this in terms of multiplicity of that zero. For example in the polynomial function -2 is a zero with multiplicity 1. However 1 is a zero with multiplicity 2. In other words it appears twice as a zero.
Example 5. Find the zeros and their multiplicities
Now let’s go backwards and start with zeros and write a polynomial function with those zeros. When working with complex number zeros, they will always appear in conjugate pairs for functions with real coefficients. In other words if 4-3 i is a zero, then 4+3 i is automatically a zero. Or if 4 i is a zero, then so is -4 i .