Polynomial Functions of Higher Degree
UNIT 6 (4) NOTES Polynomial Functions-Evaluate and Factors
Polynomial Functions of Higher Degree
Polynomial Functions All polynomial Functions : 1. Are continuous - no holes no breaks 2. Have smooth rounded curves - no points or sharp angles 3. Have degrees that are odd, even or neither 4. Have left and right end behavior
UNIT 6 (4) NOTES Polynomial Functions-Evaluate and Factors
Fundamental Theorem of Algebra Every polynomial equation over the field of complex numbers of degree higher than one has a complex solution.
Translation:
Number of Zeroes If p is a polynomial with degree
then p has exactly n zeroes (real or complex), where zeroes of multiplicity are counted m times.
Translation:
UNIT 6 (4) NOTES Polynomial Functions-Evaluate and Factors
Zeroes of Multiplicity If p is a polynomial with degree
and (x-c) occurs as a factor of p exactly m times, then c is a zero of multiplicity m.
Translation:
Irreducible Quadratic Factors If p is a polynomial with real coefficients, p can be factored into a product of linear factors (which are not necessarily distinct) and irreducible quadratic factors having real coefficients.
Translation:
UNIT 6 (4) NOTES Polynomial Functions-Evaluate and Factors
Complex Conjugates If p is a polynomial with real coefficients, complex zeroes must occur in conjugate pairs.
Translation:
Other Uses of Synthetic Division Finding the Roots and Writing Equation 1) Find all of the roots of if one of the roots is 1
UNIT 6 (4) NOTES Polynomial Functions-Evaluate and Factors
2) Find all of the roots of if one of the roots is -3
3) Find all of the roots of if two of the roots are -4 and 5
UNIT 6 (4) NOTES Polynomial Functions-Evaluate and Factors
4) Find all of the roots of f(x) if two of the roots are -2 and 4
5) Find the other x-intercepts of the function if the graph goes through (-3,0)
UNIT 6 (4) NOTES Polynomial Functions-Evaluate and Factors
Writing the Equation of Polynomials Given the Roots. 6) Write an equation of a polynomial function with integral coefficients that has a zeroes of -4 and 1. Write the answer in standard form.
7) Write an equation of a polynomial function with integral coefficients that has a zeroes of -3, -1 and 2. Write the answer in standard form.
UNIT 6 (4) NOTES Polynomial Functions-Evaluate and Factors
8) Write an equation of a polynomial function with integral coefficients that has a zero of 2+i. Write the answer in standard form.
9) Write an equation of a polynomial function with integral coefficients that has a zeroes of 0 and Write the answer in standard form.
UNIT 6 (4) NOTES Polynomial Functions-Evaluate and Factors
10) Write an equation of a polynomial function with integral coefficients that has a zeroes of 3i, 3+2i. Write the answer in standard form.
Polynomial Functions of Higher Degree
Polynomial Functions All polynomial Functions : 1. Are continuous - no holes no breaks 2. Have smooth rounded curves - no points or sharp angles 3. Have degrees that are odd, even or neither 4. Have left and right end behavior
UNIT 6 (4) NOTES Polynomial Functions-Evaluate and Factors
Fundamental Theorem of Algebra Every polynomial equation over the field of complex numbers of degree higher than one has a complex solution.
Translation:
Number of Zeroes If p is a polynomial with degree
then p has exactly n zeroes (real or complex), where zeroes of multiplicity are counted m times.
Translation:
UNIT 6 (4) NOTES Polynomial Functions-Evaluate and Factors
Zeroes of Multiplicity If p is a polynomial with degree
and (x-c) occurs as a factor of p exactly m times, then c is a zero of multiplicity m.
Translation:
Irreducible Quadratic Factors If p is a polynomial with real coefficients, p can be factored into a product of linear factors (which are not necessarily distinct) and irreducible quadratic factors having real coefficients.
Translation:
UNIT 6 (4) NOTES Polynomial Functions-Evaluate and Factors
Complex Conjugates If p is a polynomial with real coefficients, complex zeroes must occur in conjugate pairs.
Translation:
Other Uses of Synthetic Division Finding the Roots and Writing Equation 1) Find all of the roots of if one of the roots is 1
UNIT 6 (4) NOTES Polynomial Functions-Evaluate and Factors
2) Find all of the roots of if one of the roots is -3
3) Find all of the roots of if two of the roots are -4 and 5
UNIT 6 (4) NOTES Polynomial Functions-Evaluate and Factors
4) Find all of the roots of f(x) if two of the roots are -2 and 4
5) Find the other x-intercepts of the function if the graph goes through (-3,0)
UNIT 6 (4) NOTES Polynomial Functions-Evaluate and Factors
Writing the Equation of Polynomials Given the Roots. 6) Write an equation of a polynomial function with integral coefficients that has a zeroes of -4 and 1. Write the answer in standard form.
7) Write an equation of a polynomial function with integral coefficients that has a zeroes of -3, -1 and 2. Write the answer in standard form.
UNIT 6 (4) NOTES Polynomial Functions-Evaluate and Factors
8) Write an equation of a polynomial function with integral coefficients that has a zero of 2+i. Write the answer in standard form.
9) Write an equation of a polynomial function with integral coefficients that has a zeroes of 0 and Write the answer in standard form.
UNIT 6 (4) NOTES Polynomial Functions-Evaluate and Factors
10) Write an equation of a polynomial function with integral coefficients that has a zeroes of 3i, 3+2i. Write the answer in standard form.
