Theorems about roots of polynomial equations!! I need help!!

9514 1404 393

Answer:

  B (correctly marked)

  1/3, 1/2, 4

Step-by-step explanation:

The rational root theorem tells you the rational roots will be of the form:

  ±(divisor of constant)/(divisor of leading coefficient)

Here, that means they will be of the form ±{1, 2, 4}/{1, 2, 3, 6}.

Right away, you can eliminate choices A and C because there cannot be 9 in the numerator, and the number of possibilities goes beyond just 1, 2, and 4.

Possible rational roots are ...

  ±1, ±2, ±4, ±1/2, ±1/3, ±1/6, ±2/3, ±4/3

__

I like a graphing calculator for finding the roots of higher-degree polynomials. A graph of this one shows roots to be 1/3, 1/2, and 4.

_____

Descartes' rule of signs tells you there are 0 negative real roots, and 1 or 3 positive real roots. The ratio of the first two coefficients suggests that one of the roots may be relatively large, so it might work to try 4.

  ((6·4 -29)·4 +21)·4 -4 = (-20 +21)·4 -4 = 0

So, x=4 is a root. Synthetic division (2nd attachment) shows the remaining quadratic factor to be 6x^2 -5x +1. Factoring this*, we have ...

  6x^3 -29x^2 +21x -4 = (x -4)(6x^2 -5x +1) = (x -4)(3x -1)(2x -1)

The roots associated with the quadratic factor are 1/3 and 1/2.

The real roots are 1/3, 1/2, and 4.

_____

* To factor ax^2 +bx +c, you look for factors of the product ac that have a sum of b. Here, you're looking for factors of 6·1 = 6 that have a sum of -5.

  6 = (-1)(-6) = (-2)(-3) . . . sums are -7 and -5

The numbers we're looking for are -2 and -3. Then the factorization is (6x -2)(6x-3)/6 = (3x -1)(2x -1).

Note that we have identified 'p' and 'q' as the factors of 'ac' that total to 'b'. The factorization is then (ax+p)(ax+q)/a. Here, both factors end up with coefficients of x that are not unity. That is not always the case.

There are other ways to make use of 'p' and 'q'. You can also rewrite the quadratic as ax^2 +px +qx +c, and factor by grouping.

What is the difference between a continuous or a discrete function? Explain.

Find the equation of the line passing through the points (5,21) and (-5,-29)

-2 1/2 divided by 3 1/2 ?

I need help on this question ​

Select the correct answer.
How does Gary's character reflect the passage's theme?
A.
The character's dialogue reveals that heroes are people of very few words.
B.
The character demonstrates that all kids are afraid of being caught skipping class.
C.
The character illustrates that bullies do not care whom they push around.
D.
The character's transformation shows that the unlikeliest of people can be heroes.

What is the slope,m, of the table?

What is the y-intercept,b, of the table?

What is the equation that represents the table?

You should see the picture and I don’t want to type it out so just look at it and there the question. Also I’m giving brainist

Helpp me pls for this one

A family wanted to stay in a hotel for 4
days. The hotel charges $60 a night. What is the
total they will pay?

Anyone know how to do this? pls explain how

PLS HELP
solve the system of equation using elimination:
3x+y=-2
5x-y=-10

Rearrange the area formula to solve for |
A=lw

Brandi earned $140 from a summer job to buy new clothes for school. She found several items she would like to buy, but is trying to decide if she has enough money to buy all the items she likes. She found two pairs of shorts for $25 each and four shirts with an average cost of $19 per shirt. She will have to pay 7.5% sales tax.


PART A: If she buys all of the items, how much sales tax will she have to pay?

The sales tax for Brandi's purchase will be $(insert answer here)


PART B: What is the total cost of her purchase including sales tax?

Brandi will spend $(insert answer here) on her purchases, including sales tax.

What is the answer to 3-(-8)

i need help with this 7th grade

What is an equation of the line whose graph is shown? A. y = -x B. y = 2x C. y = x D. y = 2

42 + (3 x 6 - 7
PEMDAS

An assembly was held for the 7th graders. When first entering the gym 5 out of 8 of the 7th grade boys had on hoodies. If 320 boys attend the assembly , what is a reasonable amount of them that will be wearing hoodies?
140
200
180
160

Determine if the two triangles are congruent. If they are, state how you know.
A) Not enough information
B) SAS
C) ASA
D) SSS

Which number line shows the graph of -2.75 -3
-2
-1
0
1
2
3
0
1
2.
3
-3 -2
-1
0
1
2 3
-3 -2
0
1
2.
3

For the following quadratic equation, find the discriminant.
x^2+ 10x + 44 = 0

Julio pald 1.2 times his normal hourly rate for each hour he works over 30 hours a week Last week he worked 43 hours and earned $574.56

I cant seem to understand this. Please, someone help :)

Is this isosceles triangle acute or equiangular?

if any of you guys know someone named shurmpy or molly plz tell me they real good freinds to me

HELP PLS BRAINLIEST FOR CORRECT ANSWER

select the cost of 2 pounds of blueberries

$2

$4.50

$9

$14.00

somebody pleaseee help it’s due today

What is the greatest common factor of 28 and 49

To control an infection, a doctor recommends that a patient who weighs 92 pounds be given 320 milligrams of antibiotic. If the antibiotic is given proportionally according to the patient's weight, how much antibiotic should be given to a patient who weighs 138 pounds?
400 milligrams
480 milligrams
550 milligrams
600 milligrams