Roots and Multiplicity
A root of odd multiplicity does this around the x-axis.
What is cross?
Polynomials start in this quadrant if the leading coefficient is positive and the degree is odd.
What is the third quadrant?
x^2+1 has what type of roots? A complete description is needed.
What are complex conjugate roots?
This theorem guarantees the existence of a solution to a polynomial in the complex number system.
What is the Fundamental Theorem of Algebra
The term used when a polynomial is divided by a binomial and has a remainder of zero.
What is a factor?
A root of even multiplicity does this around the x-axis.
What is bounce?
Even-degree polynomials end in the fourth quadrant if the leading coefficient is this type of number.
What is a negative real number?
The fundamental theorem of algebra allows to conclude that P(x) has solutions in this number system.
What is the complex number system?
The theorem that states the number of positive real solutions is the number of sign changes less an even integer, and the number of negative real solutions is the number of sign changes in f(-x) less an even integer.
What is Descartes' Rule of Sign?
The remainder when P(x) = x^2 + 2x - 3 is divided by x-1 .
What is zero?
Suppose a polynomial has (x-4)^3 as a factor. Tell the root and the multiplicity.
What is 4 with a multiplicity of 3?
Ending behavior for
2x^3-4x^2+2x-9
What is Quadrant 1?
If we have an integer coefficient polynomial, and we know that 1+sqrt(2) is a solution, what must another solution be?
What is
1-sqrt(2)?
This theorem states that if an integer polynomial has rational roots, they must be factors of frac{p}{q} where q is a factor of the leading coefficient and p is a factor of the constant term.
What is the Rational Zero Theorem?
A factor of x^3 - 4x^2 + x+6 if x = 3 is a root.
What is x-3?
The polynomial with these characteristics:
1) Degree 4
2) Roots at -1 and 2
3) cross at -1 and cross at 2, with 2 having a higher multiplicity than -1.
4) Polynomial is monic.
What is
P(x) = (x+1)(x-2)^3
The quadrant where -4x^4-7x^5+2 starts its graph.
What is quadrant 2?
True or False, if a real coefficient polynomial has
1+i as a root, then another root must be -1-i .
What is False?
This theorem suggests that for the polynomial, P(x) = 2x^2-4x+1 the real roots are all within the interval (-3,3)
What is the Cauchy Bounds Theorem?
The quotient of (x^3-4x^2+x+6)/(x-3).
What is x^2-x-2?
Complete description of roots for P(x) = -2x(x-1)^2(x+2)^3
What is: roots are 0 1, -2. Multiplicities are 1,2,3 respectively. The graph crosses at x=0 and x=-2 and bounces at x=1?
A degree-two polynomial with a positive leading coefficient has this end behavior when divided by a factor (x-r), the remainder is zero.
What is starting in quadrant 3 and ending in quadrant 1?
An integer coefficient polynomial has one root equal to
1+i and another root x=3 of multiplicity 2, the factored form is this.
What is (x-3)^2 (x-(1+i))(x-(1-i))?
This theorem suggests that if a function is continuous and has a sign change over an interval [a,b], there must at least one solution in (a,b).
What is the Intermediate Value Theorem?
The standard form of a polynomial whose remainder when dividing by (x-1), (x-2), and (x+3) is zero.
What is
x^3-7x-6?