Polynomial Equations and Inequalities

What are zeros?

A zero (aka root) of a polynomial function is the value of x such that f(x)=0. In other words roots (solutions, or zeros) are the x-intercepts. *If zero is given take the opposite sign for that root. Example: zeros: 1, -2 (x-1)(x-(-2)) *If a zero is 1-i,apply the Conjugate Pairs Theorem: (x-(1-i))(x-(1+i))

What are the Domain & Range symbols and their meanings?

( ) means that it cannot include the number [ ] means that it can include the number U means union or and

What are some general equations for multiplying and simplifying?

(A-B)(A+B) + A^2 - B^2 (A-B)^2 = A^2 - 2AB + B^2

What is a binomial?

A binomial is a sum x+y where x and y represent numbers. If n is a positive integer, then a general formula for expanding (x+y)^n is given by the binomial theorem. 1. The following formula is true for each of the first n terms of the expansion: (coefficient of term)(exponent of a)/(number of term) = coefficient of next term 2. The following formulas are valid for (a+b)^n for an arbitrary positive integer n. Coefficient of the (k+1)st term in the expansion of (a+b)^n (n k)=C(n,k)=n!/k!(n−k)!, k=0,1,2,3,....n Binomial Theorem

What is the Factor Theorem?

A polynomial P(x) has a factor (x−c) if and only if f(c)=0.

What is Descartes' Rule of Signs?

A polynomial can have positive, negative, or non real roots. Since any nth degree polynomial equation has exactly n roots, the minimum number of complex roots is equal to (n−(p+q)). Where p denotes the maximum number of positive roots, and q denotes the maximum number of negative roots (both of which can be found out using Descarte's rule of sign), and n denotes the degree of the equation.

What does the Quadratic Formula give you?

A quadratic equation with real or complex coefficients has 2 solutions (aka roots). These 2 solutions may or may not be distinct or real.

What is completing the square?

Completing the square is taking the Quadratic Formula and re-writing it in a form that completes the square. ax^2+bx+c => a(x+h)^2+k where h = -b/2a & k = c-b^2/4a

What is Zero of multiplicity k?

If (x−r)^k is a factor of a polynomial function f and (x−r)^(k+1) is not a factor of f, then r is called a zero of multiplicity of k. A polynomial function with the highest exponent, depending on the number of times that factor occurs in the product, the exponent on the factor that the zero is a solution for, gives the multiplicity of that zero. *The exponent indicates how many times that factor would be written out in the product, this gives us a multiplicity.

What is the Complete Factorization Theorem for Polynomials?

If P(x) is a polynomial of degree n>0, then there exist n complex numbers, c1,cc,...,cn such that P(x)=a(x−c1)(x−c2)...(x−cn).

What is the Fundamental Theorem of Algebra?

If a polynomial P(x) has positive degree and complex coefficients, then P(x) has at least one complex zero.

What is the Conjugate Pairs Theorem?

If a polynomial P(x) of degree n>1 has real coefficients and if z=a+bi with b≠0 is a complex zero of P(x), then the conjugate z¯=a−bi is also a zero of P(x). Example: 2-3i is a zero of p(x)=x^3-(3x)^2+9x+13 as shown here: p(2-3i) =(2-3i)^3-3(2-3i)^2+9(2-3i)+13 =(-46-9i)-3(-5-12i)+(18-27i)+13 =-46-9i+15+36i+18-27i+13 =0 By the conjugate pair theorem, 2+3i is also a zero of p(x). p(2+3i) =(2+3i)^3-3(2+3i)^2+9(2+3i)+13 =(-46+9i)-3(-5+12i)+(18+27i)+13 =-46+9i+15-36i+18+27i+13 =0.

How do you carry of the Division of Polynomials?

If f(x) and P(x) are polynomials and if P(x)≠0, then there exists unique polynomials q(x) and r(x) such that f(x)=P(x)?q(x)+r(x),Where either r(x)=0 or the degree of r(x)is less than the degree of P(x). The polynomial q(x) is the quotient, and r(x) is the reminder in the division of f(x)by P(x).

What are Rational Zeros of Polynomials?

If the polynomial has integer coefficients and if c/d is a rational zero of P(x) such that c and d have no common prime factor, then 1. the numerator c of the zero is a factor of the constant term a0 2. the denominator d of the zero is a factor of the leading term an Example: P(x) = 3x^3 - 2x^2 -7 x - 2 P : c/d factors of constant term of poly/factors of leading coefficient P : c/d -2/3 : +1, -1, +2, -2/ +1, -1, +3, -3 P(-1) = 3(-1)^3 - 2(-1)^2 - 7(-1) - 2 = -3-2+5 = 0 -1 is a factor of P aka (x-(-1)) Factor Theorem: P(x) = (x+1)Q(x) Q(x) = (x+1)/(3x^3 - 2x^2 -7 x - 2) = 3x^2 - 5x - 2 P(x) = (x+1)(3x^2 - 5x - 2) * Factor Q(x) to get the remaining zeros of x (3x^2 - 5x - 2) => (3x+1)(x-2) Zeros: (x+1)(3x+1)(x-2) x = -1, -1/3, 2

What is Descartes' Rule of Signs (Negative Roots)?

Negative roots As a corollary of the rule, the number of negative roots is the number of negative integers after negating the coefficients of odd-power terms (otherwise seen as substituting the negation of the variable for the variable itself), - or fewer than it by a multiple of 2. Example: −x^3 + x^2 + x − 1 This polynomial has two sign changes, has two or zero positive roots. The factorization of this polynomial is (x−1)^2(x+1). So here, the roots are 1 (twice) and −1.

What is Descartes' Rule of Signs (Positive Roots)?

Positive roots The rule states that if the terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either - equal to the number of sign differences between consecutive nonzero coefficients, - or is less than it by a multiple of 2. * Multiple roots of the same value are counted separately. Example: x^3 + x^2 − x − 2 This has one sign change between the second and third terms. Therefore it has exactly one positive root. * Note that the leading sign needs to be considered although it doesn't affect the answer in this case. In fact, this polynomial factors as x^3 + x^2 − x − 2 so the roots are −1 (twice) and 1.

What does the graphing of a Linear Inequality represent?

When graphed, a linear inequality represents a region, or half plane, which contains solutions that will make the statement true.

What is Range/

Range is the possible y values of a function that we get when we substitute all possible x-values into the function. When finding the Range, remember: * substitute different x-values into the expression for y to see what is happening * look for min & max values of y * draw a sketch

What does the Discriminant determine?

The Discriminant is the part of the Quadratic Formula: b^2 - 4ac (its symbol is a triangle) The Discriminant determines the number and nature of the roots. There are 3 cases: b^2 - 4ac > 0 => 2 real solutions b^2 - 4ac = 0 => 1 real solution b^2 - 4ac < 0 => no real solutions, 2 (non-real) complex solutions

What is Domain?

The domain is a set of "input" or values for the function. The function provides an "output" or value for each member of the Domain. When finding the Domain, remember: * the denominator cannot equal zero * values under the square root must be positive

How does the use of the Summation Notation affect the Binomial Theorem?

Using the summation notation, we may write the binomial theorem (a+b)^n = n∑k=0 (n k) (a^n−k)(b^k) *There are (n+1)(not n terms) terms in the expansion of (a+b)^n, and so (n k) (a^n−k)(b^k) is a formula for the (k+1)st term of the expansion.