Polynomial Functions and equations

1. Lesson 4: Remainder and Factor Theorems

1.1. If a polynomial f(x) is divided by (x-k), the remainder is f(k)

1.2. Factor Theorem: k is a zero of a polynomial if (x-k) is a factor.

2. Lesson 5: Rational Zeros and Fundamental Theorems of Algebra

2.1. If f(x)=anxn+…+a1x+a0 has integer coefficients, then every rational zero of f has the following form: p/q factors of constant term /factors of the leading coefficients

2.2. If a zero appears more than once, it has a multiplicity

3. Lesson 6: Descartes rule of signs

3.1. The number of positive real zeros of f is equal to the number of changes in sign of the coefficients of f(x) or is less than this by an even number

3.2. The number of negative real zeros of f is equal to the number of changes of the coefficients of f(-x) or is less than this by an even number

4. Lesson 7: Complex and irrational conjugates

4.1. If f is a polynomial function with rational coefficients, and a and b are rational numbers such that is irrational. If a+ is a zero of f, then a- is also a zero of f

5. Lesson 1: Factor and Solve Polynomial equations

5.1. Sum of Two Cubes: a³+b³=(a+b)(a²-ab+b²)

5.1.1. Difference of Two Cubes: a³-b³=(a+b)(a²+ab+b²)

5.1.2. Difference of Two Cubes: a³-b³=(a+b)(a²+ab+b²)

5.1.3. Difference of Two Cubes: a³-b³=(a+b)(a²+ab+b²)

5.2. Difference of Two Cubes: a³-b³=(a+b)(a²+ab+b²)

5.3. Signs: "Same- opposite- positive"

6. Lesson 2: Long division of Polynomials

6.1. The degree of the remainder must be less than the degree of the divisor

6.2. When you divide a polynomial P(x) by a divisor D(x), you get a quotient polynomial Q(x) and a remainder polynomial R(x)

7. Lesson 3: Synthetic Division

7.1. a quick way to divide polynomials

7.2. used when the divisor is in the form "x-c"

7.3. steps- 1. Write the appropriate coefficients to represent the dividend and divisor 2. Add going down and multiply going down 3. The last term is the remainder and previous terms are the coefficients of the quotient 4. the degree of the quotient will be one less

8. Lesson 8: Graphing Polynomial Functions

8.1. Factor the equation, find the roots

8.2. To find the y-intercept plug in 0 for x

8.3. Consider the end behavior at each zero: Will it cross (multiplicity of one) or bounce (multiplicity two)?

8.4. Remember! Imaginary solutions do not appear on the graph!

9. Lesson 9: Applications of polynomials

9.1. Independent and Dependent Variables: The independent variables will lie on the y-axis while the dependent variables lie on the x-axis.

9.2. "if x equals a number, what does y equal?", "if y equals a number, what does x equal?" and dimensions of an object.