MTH 161 Ch2 Polynomials Zeros and Multiplicity (2.2)

How do you classify a polynomial by degree?

Degree is the highest exponent: 0 constant, 1 linear, 2 quadratic, 3 cubic, 4 quartic, 5 quintic.

How do you classify a polynomial by number of terms?

One term: monomial. Two: binomial. Three: trinomial. More: polynomial.

What do polynomial graphs look like?

Continuous: no breaks, holes, or sharp corners.

End behavior: even degree, positive leading coefficient?

Both ends go UP.

End behavior: even degree, negative leading coefficient?

Both ends go DOWN.

End behavior: odd degree, positive leading coefficient?

Down on the left, up on the right.

End behavior: odd degree, negative leading coefficient?

Up on the left, down on the right.

What are zeros, x-intercepts, solutions and roots?

All the same idea: the x-values where f(x) = 0. Set each factor equal to zero to find them.

What is multiplicity?

The number of times a factor is repeated, the exponent on that linear factor.

Odd vs even multiplicity at a zero?

Odd multiplicity: the graph CROSSES the x-axis. Even multiplicity: the graph BOUNCES (tangent) off the x-axis.

For x^3 (x+4)^2, what happens at each zero?

x = 0 has multiplicity 3 so it crosses. x = -4 has multiplicity 2 so it bounces.

How many real zeros can a polynomial have?

The degree is the max number of real zeros. Even degree has an even number of real zeros and odd degree has an odd number, counting multiplicity and ignoring complex roots.

Maximum number of turning points?

A degree n polynomial has at most n - 1 turning points.

How do you find the y-intercept of a polynomial?

Compute f(0), which is the constant term.

Factor x^3 + 3x^2 - 10x and list the zeros.

x(x+5)(x-2). Zeros at x = 0, -5, 2.

Factor 2x^3 - 5x^2 - 3x and list the zeros.

x(2x+1)(x-3). Zeros at x = 0, -1/2, 3.

Factor (x^2 - 4)(3x - 1) and list the zeros.

x = 2, x = -2, x = 1/3.

What does the Intermediate Value Theorem say?

If f(a) and f(b) have opposite signs, there is guaranteed to be a zero between a and b. If they have the same sign you cannot conclude anything.

IVT example with f(x) = x^3 - 13x^2 + 33x + 35?

f(-1) = -12 and f(0) = 35 have opposite signs, so there is a zero between -1 and 0.

What will the tests ask about polynomials?

End behavior, zeros, multiplicity, and cross or bounce at each zero. Hand graphing is not expected. Descartes Rule of Signs and the Rational Zero Theorem are introduced but not tested.

What is the symmetry rule for polynomials?

Only even powers means even function (y-axis symmetry). Only odd powers means odd function (origin symmetry).