Factor, Write & Graph Polynomials

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x=a is a zero of f(x)

x=a is a solution of f(x)=0

(x-a) is a factor of f(x)

(a,0) is a x-intercept

NEVER FACTORABLE: (a^3 +/- b^3)= (a +/- b)(a^2 -/+ ab +b^2)

Quadratic Like Trinomial: 2nd term is half the value of the 1st term. Solve as if factoring a quadratic, square x values

Repeated Zeros: multiplicity

A factor (x-a)^k yields a repeated zero x=a of multiplicity k

If k is odd, graph crosses x-axis at x=a

If k is even, graph touches x-axis at x=0

Leading Coefficient Test: coefficient of the leading term, indicates end behavior of a polynomial

-odd degree: flips/opposite end behavior

-even degree: same end behavior

-negative coefficient: right end: negative

-positive coefficient: right end: positive

Turning Point Test: turning point is the maximum or minimum where the polynomial changes direction

-odd degree: even number of turning points

-even degree: odd number of turning points

A polynomial with an n degree has at most (n-1) turning points

Intermediate Value Theorem: If two points are given with one being positive and the other negative, there must be a real zero.

When writing equations of polynomials, ensure that rationals are not part of the term. (x-1/2)->(2x-1). Also ensure that the equation is in standard form. Work backwards and include multiplicity when needed. A GCF can help create a y-int.

To Create Proper Polynomial Graphs:

1. Apply the LCT

2. Find the x-int(s), check for mult.

3. find y-int

4. Use a graphing calculator to find additional points halfway b/n the x-intercepts

5. sketch a smooth, continuous curve

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Polynomials Unit 3.3