Question

In: Math

a) For the following polynomial; a. Use the Rational Zero Test to list all possible rational...

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

Solutions

Expert Solution

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)


Related Solutions

Use the Rational Zero Test to list all possible rational zeros of f. Test each possible...
Use the Rational Zero Test to list all possible rational zeros of f. Test each possible rational zero to determine whether it is an actual zero of f. (Enter your answers as a comma-separated list. If there are no rational zeros, enter NONE.) f(x) = x4 + 2x3 + 4x2 + 3x − 10 possible rational zeroes x= actual rational zeroes x=
Find all zeros of the polynomial function. Use the Rational Zero​ Theorem, Descartes's Rule of​ Signs,...
Find all zeros of the polynomial function. Use the Rational Zero​ Theorem, Descartes's Rule of​ Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. f(x)=2x4-17x3+13x2+53x+21 The zeros of the function are?
Find all zeros of the polynomial function. Use the Rational Zero Theorem, Decarte's Ruke of Signs,...
Find all zeros of the polynomial function. Use the Rational Zero Theorem, Decarte's Ruke of Signs, and possibly the graph of the polynomial function showny by a graphing utility as ab aid in obtaining the first sero or the firtst root. f(x)=x^4-4x^3-x^2+14x+10 Find the zeros of the polynomial function.
x^3-34x+12=0 1) list all rational roots that are possible according to the Rational Zero Theorm a)...
x^3-34x+12=0 1) list all rational roots that are possible according to the Rational Zero Theorm a) +-1 b)+-1,+-2,+-3,+-4,+-6,+-12 c)+-12 d) 1,2,3,4,6,12 b)use synthetic division to test several possible rational roots in order to identify one actual root c) use the root from part b to solve the equation the solution set it?
1. List the possible rational zeros of the the polynomial function. Do not find the zeros....
1. List the possible rational zeros of the the polynomial function. Do not find the zeros. f ( x ) = 3 x 5 − 5 x 2 + 6 x − 1 A) ± 1 , ± 3 B) ± 1 , ± 1 3 C) ± 1 , ± 1 3 , ± 3 D) ± 3 , ± 1 3 Group of answer choices A B C D 2. Find zeros and write the polynomial in factored...
List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see...
List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see which actually are zeros). (Enter your answers as a comma-separated list.) Q(x) = x4 − 5x3 − 3x + 6
use the Rational Zeros Theorem to find all the real zeros of each polynomial function given...
use the Rational Zeros Theorem to find all the real zeros of each polynomial function given below: f(X)=3X^3 + 6X^2 -15X - 30 f(X) = 2X^4 - X^3 - 5X^2 + 2X + 2
Given: Polynomial P(x) of degree 6 Given: x=3 is a zero for the Polynomial above List...
Given: Polynomial P(x) of degree 6 Given: x=3 is a zero for the Polynomial above List all combinations of real and complex zeros, but do not consider multiplicity for the zeros.
All polynomial functions are continuous over all real numbers true of false rational functions are discontinuous...
All polynomial functions are continuous over all real numbers true of false rational functions are discontinuous where the numerator is equal to zero true or false exponential functions are continuous over all real numbers true or false log functions are continuous over all real numbers true or false The first derivative of a function gives the average rate of change at a point true or false The second derivative of a function gives the instantaneous rate of change at a...
1.Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever...
1.Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the Quadratic Formula, or other factoring techniques. (Enter your answers as comma-separated lists. Enter all answers including repetitions. If an answer does not exist, enter DNE.) P(x) = 12x4 − 11x3 − 18x2 + 5x rational zeros     x = 2. Find all rational zeros of the polynomial. (Enter...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT