Question

a) For the following polynomial; a. Use the Rational Zero Test to list all possible rational roots b. Use Descartes Rule of Signs to provide the possible numbers of positive and negative real roots c. Factor the polynomial completely. ? 3 + 4? 2 + 9? + 36

b) For the following polynomial; d. Use the Rational Zero Test to list all possible rational roots e. Use Descartes Rule of Signs to provide the possible numbers of positive and negative real roots f. Factor the polynomial completely. ? 4 + 3? 3 − 7? 2 − 27? − 18

Answer 1

1) For the following polynomial:

a. Use the Rational Zero Test to list all possible rational roots.?^3 + 4?^ 2 + 9? + 36.

p: all possible factors of the constant

q: all possible factors of the leading coefficient

p/q: all possible rational roots

p (all factors of 36) : ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36

q (all factors of 1) : ±1

Possible Rational Roots (p/q) : ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36

b. Use Descartes Rule of Signs to provide the possible numbers of positive and negative real roots. ?^3 + 4?^ 2 + 9? + 36

# of positive real roots = # of sign changes of f(x) & less than that by an even number.

# of negative real roots = # of sign changes of f(-x) & less than that by an even number.

f(x) = ?^3 + 4?^ 2 + 9? + 36

# of positive real roots = # of sign changes of f(x) = 0

f(-x) = -?^3 + 4?^ 2 - 9? + 36

# of negative real roots = # of sign changes of f(-x) = 3 or 1

Total zeros depends on the degree, so here it is 3.

Imaginary zeros always appear in even numbers.

c. Factor the polynomial completely. ?^3 + 4?^ 2 + 9? + 36.

?^3 + 4?^ 2 + 9? + 36

Factoring by grouping works,

?^3 + 4?^ 2 + 9? + 36

x^2 (x+4) + 9 (x+4)

(x+4) (x^2 + 9) : This is the factored form.

Further factoring it in terms of complex roots,

(x+4) (x+3i) (x-3i)