Chapter 3

Question 1

Real Zeros of Polynomials

Answer 1

If p is a polynomial and c is a real number, then c is a zero of p, x=c is a solution when p(x)=0, x-c is a factor of p(x) and x=c is an x intercept on the graph of p

Question 2

Intermediate value thm for polynomials

Answer 2

If p is a poly. fcn and p(a) and p(b) have opposite signs, there exists at least one value c between a and b for which p(c)=0

Question 3

Remainder thm

Answer 3

If p(x) is divided by x-c, then the remainder is the value of p(c)

Question 4

Factor thm

Answer 4

C is a zero of p if and only if x-c is a factor of p(x)

Question 5

Rational zeros thm

Answer 5

P/q
P is factor of constant coefficients
Q is a factor of the leading coefficient

Question 6

Upper and lower bounds thm

Answer 6

If divide p(x) by x-b(b>0) using synthetic, and if the row contains only positive quotient and remainder, then b is an upper bound for zeros of p.

If divide p(x) by x-a( with a<0) using synthetic, and if row contains quotient and remainder with alternating signs, then a is a lower bound for zeros of p.

Question 7

Complete factorization thm

Answer 7

P(x) is poly. degree n>=1, then exist complex numbers a (a doesn’t equal zero) such that p(x)= a(x-c1)(x-c2)…(x-cn)

Question 8

Zeros thm

Answer 8

Every Poly of degree n>= 1 has n zeros, provided a zero of multiplicity k is counted k times