FUNDAMENTAL THEOREM OF ALGEBRA AND FINDING REAL ROOTS
FUNDAMENTAL THEOREM OF ALGEBRA AND FINDING REAL ROOTS Honors Advanced Algebra Presentation 2 -6
WARM-UP Given the roots, write the factors of the quadratic and the polynomial. 1. x = -3, 2 2. x = 4, 3 3. x = -5, 5 4. x = 1, 1
ROOTS OF A POLYNOMIAL The places where a polynomial crosses or touches the x-axis are called the roots of the polynomial. They are also known as x-intercepts, zeros, or solutions. Roots can be found by setting the polynomial equal to 0 and solving.
THE FUNDAMENTAL THEOREM OF ALGEBRA
MULTIPLICITY OF A ROOT A root can occur once or multiple times. If the root repeats, the number of times is known as the multiplicity of a root. If the multiplicity is even, the graph will touch the xaxis but not cross it; if the multiplicity is odd, the graph will intersect the x-axis. Example: Write a polynomial with roots 1 with a multiplicity of 2 and -3 with a multiplicity of 1.
MULTIPLICITY OF A ROOT Multiplicity of 1 Multiplicity of 2 Multiplicity of 3
MULTIPLICITY OF A ROOT Example 2: Write a possible polynomial as the product of factors given the graph below.
RATIONAL ROOTS THEOREM
RATIONAL ROOTS THEOREM
RATIONAL ROOTS THEOREM
IRRATIONAL ROOTS THEOREM
IRRATIONAL ROOTS THEOREM
COMPLEX CONJUGATE ROOT THEOREM If a + bi is a root of a polynomial equation with realnumber coefficients, then a – bi is also a root. Imaginary roots always come in conjugate pairs. Example: If 3 + 2 i is a root of a polynomial, what is another root of the polynomial? What is the corresponding factor?
EXAMPLE OF FINDING ROOTS
EXAMPLE OF FINDING ROOTS
EXAMPLE OF FINDING ROOTS
EXAMPLE OF WRITING A POLYNOMIAL FUNCTION GIVEN ZEROS
HOMEWORK P. 121, #24 -25, 30 -35 Pg. 127 -128, #12 -32 even, 38 -43
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