Polynomial Equations and Inequalities

Mai multe ca aceasta

MAT.126 P.1

după David Kedrowski

Polynomials

după Celina Yang

MAT.126 4.4

după David Kedrowski

Romantic Elements in Frankenstein

după Bhatt Nisarg

Polynomial Equations and Inequalities

Polynomial Equations

Factor Theorem

(x-b) is a factor of a polynomial P(X) if and only if P(b) = 0. Similarly, (ax-b) is a factor of P(x) if and only if P(b/a) = 0 Example: To determine if x-3 is a factor of P(x)=x³ - x² - 14x + 24, calculate P(3): P(3) = (3)³- (3)² + 14(3) + 24 = 27 – 9 – 42 + 24

Rational Zero Theorem

Suppose P(x) is a polynomial function with integer coefficients and x= b/a is a zero of P(x), where a and b are integer and a≠0. Then, b is a factor of the constant term of P(x) a is a factor of the leading coefficient of P(x) ax-b is a factor of P(x) Example: Consider a factorable polynomial such as P(x) = 3x² + 7x – 2. Since the leading coefficient is 3, one of the factors must be of the form (3x-b), where b is a factor of the constant term 2 and P(b/3) = 0 To determine the values of x that should be tested to find b, the integral zero theorem is extended to include polynomials with leading coefficients that are not one.

Integral Zero Theorem

If (x-b) is a factor of a polynomial function P(x) with leading coefficient 1 and remaining coefficients that are integers, then b is a factor of the constant term of P(x). Example: Consider the polynomial P(x) = x² + 6x – 5 A value that satisfies P(b) = 0 also satisfies b² + 6b – 5 =0, or b² + 6b = 5. Since the product of b² + 6b – 5 is 5, the possible integer values for the factors in the product are the factors of 5. They are ±1,±5.

Remainder Theorem

The remainder theorem can also be used to find the point (x,y)

P(x)/(x-b)=Q(x)+R/(x-b) Q(x)=P(x)/(x-b)-R/(x-b) Q(x)=(P(x)-R)/(x-b) Therefore: [x,P(x) ],(b,R)

The remainder theorem states that when a polynomial function P(x) is divided by x-b, the remainder is P(b); and when it is divided by ax-b, the remainder is P(b/a), where a and b are integers, and a≠0

[Q(x) ](ax-b)+R=P(x) [Q(b/a) ][a(b/a)-b]+R=P(b/a) Q(b/a)(0)+R=P(b/a) R=P(b/a)

[Q(x) ](x-b)+R=P(x) [Q(b) ](b-b)+R=P(b) R=P(b)

Families of Polynomial Function

Solving Inequalities

Pre-requisite Skills

[Q(x) ](x-b)+R=P(x)

Corresponding Statement

(P(x))/(x-b)=Q(x)+R/(x-b) ,x≠b

Quotient Form

Q(x) (x-b)√(P(x) ¦_ R

LONG DIVISION FORM

Q(x) = Quotient

P(x) = Polynomial equation/ Dividend

R = Remainder

(x-b) = Divisor/Binomial equation