Unit 3: Polynomials

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Unit 3: Polynomials by Mind Map: Unit 3: Polynomials

1. Solving Equations/Inequalities

1.1. Factoring

1.1.1. ALWAYS check for a GCF

1.1.2. Grouping (4 terms)

1.1.3. Difference of Squares

1.1.3.1. (a^2-b^2) = (a+b)(a-b)

1.1.4. Sum of Cubes

1.1.4.1. (a^3+b^3) = (a+b)(a^2 - ab + b^2)

1.1.5. Difference of Cubes

1.1.5.1. (a^3-b^3) = (a-b)(a^2 + ab + b^2)

1.2. Completing the Square (not covered in M3)

1.3. Quadratic Formula

1.3.1. Complex Roots (see roots bubble)

1.4. Solving Inequalities

1.4.1. f(x) > 0 is true when ______

1.4.2. f(x) < 0 is true when ______

2. Roots/Zeros

2.1. "Regular" Root

2.2. Double Root

2.3. Wiggle Root (Triple Root)

2.4. "Multiplicity" of Roots

2.4.1. What does it mean to have a "multiplicity?"

2.5. Imaginary Root

2.5.1. Conjugates. What does it mean to be a "conjugate?"

2.6. How do these roots look like?

2.7. Rational Root Theorem

2.7.1. A degree with polynomial n can have at most n real roots.... how can we find these roots using the Rational Root Theorem?

2.7.2. We can use what kind of division technique to find the roots (given we use Rational Root Theorem first)?

2.8. The Fundamental Theorem of Algebra

2.8.1. What does this say?

3. Graphing/Sketching

3.1. Even/Odd Degree

3.1.1. What does an odd degree mean? What does it mean to be "odd?"

3.1.2. What does an even degree mean? What does it mean to be "even?"

3.2. Leading Coefficient +/-

3.2.1. What does a negative leading coefficient mean?

3.2.2. What does a positive leading coefficient mean?

3.2.3. Why does this matter?

3.3. End Behavior

3.3.1. Be able to describe the end behaviors related to the degree and leading coefficient. Think: Make a table!

3.3.1.1. Even Degree Positive Leading Coefficient: both x= +/- ∞, f(x) tends to +∞

3.3.1.2. Even Degree Negative Leading Coefficient: both x=+/- ∞, f(x) tends to -∞

3.3.1.3. Odd Degree Positive Leading Coefficient: As x= +∞, f(x) tends to +∞. Vice versa: As x=-∞, f(x) tends to -∞

3.3.1.4. Odd Degree Negative Leading Coefficient: As x= +∞, f(x) tends to -∞. Vice versa: As x=-∞, f(x) tends to +∞

3.3.1.5. Be able to sketch using the end behavior and make examples (generalized) for all four types of end behaviors. Example: What does the end behavior of a 17th degree polynomial with a negative leading coefficient? What about 6th degree polynomial with a positive leading coefficient?

3.4. Extrema

3.4.1. What can you tell about a graph when you look for the total amount of extrema?

3.5. Roots! (see roots bubble)

3.5.1. Emphasis on what the root is (regular, double, wiggle, imaginary)

3.5.1.1. Extend: How many possible solutions can there be if it's a quartic function (x^4)?

3.5.1.1.1. 0: 2 pairs of conjugate roots

3.5.1.1.2. 1: 1 real solution (how?)

3.5.1.1.3. 2: 2 real solutions (how?)

3.5.1.1.4. 3: 3 real solutions (how?)

3.5.1.1.5. 4: 4 real solutions (how?)

3.5.1.1.6. Using the examples above, can you make equations out of each scenario? For example: 0 real roots would be (x^2+1)(x^2+2) (why?)

3.5.1.1.7. What do you notice? What do you wonder?

3.5.1.2. Extend: How many possible solutions can there be if it's a quintic function (x^5)? List the possible outcomes.

3.5.1.2.1. Use x^4 as an example and make your conjectures!

3.5.2. A polynomial of degree "n" has how many possible roots? See also: Rational Root Theorem and Fundamental Theorem of Algebra

3.6. Minimum Degree of Polynomial

3.6.1. See Roots bubble above. If the graph has 2 roots, does it necessarily mean it's a quadratic?

4. Operations

4.1. Adding/Subtracting/Multiplying (not covered in M3)

4.2. Dividing

4.2.1. Long Division

4.2.1.1. Careful! Be sure if you have missing powers, you have to do what?

4.2.2. Synthetic Division

4.2.2.1. Careful! Be sure if you have missing powers, you have to do what?

4.2.3. Remainders

4.2.3.1. Remainder Theorem

4.2.3.1.1. What does this theorem say? What kind of division do we have to use?

4.2.3.2. How do you write remainders when you divide polynomials?

4.2.4. Factor Theorem

4.2.4.1. What does this theorem say?

5. Writing Equations

5.1. Sum and Product of Roots

5.2. Complex Solutions

5.3. How do you use roots to write polynomial equations?

6. What is a Polynomial Function?