1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
AVprozaik [17]
2 years ago
13

Find a polynomial $f(x)$ of degree $5$ such that both of these properties hold: $\bullet$ $f(x)$ is divisible by $x^3$. $\bullet

$ $f(x) 2$ is divisible by $(x 1)^3$. Write your answer in expanded form (that is, do not factor $f(x)$).
Mathematics
1 answer:
frosja888 [35]2 years ago
6 0

There seems to be one character missing. But I gather that <em>f(x)</em> needs to satisfy

• x^3 divides f(x)

• (x-1)^3 divides f(x)^2

I'll also assume <em>f(x)</em> is monic, meaning the coefficient of the leading term is 1, or

f(x) = x^5 + \cdots

Since x^3 divides f(x), and

f(x) = x^3 p(x)

where p(x) is degree-2, and we can write it as

f(x)=x^3 (x^2+ax+b)

Now, we have

f(x)^2 = \left(x^3p(x)\right)^2 = x^6 p(x)^2

so if (x - 1)^3 divides f(x)^2, then p(x) is degree-2, so p(x)^2 is degree-4, and we can write

p(x)^2 = (x-1)^3 q(x)

where q(x) is degree-1.

Expanding the left side gives

p(x)^2 = x^4 + 2ax^3 + (a^2+2b)x^2 + 2abx + b^2

and dividing by (x-1)^3 leaves no remainder. If we actually compute the quotient, we wind up with

\dfrac{p(x)^2}{(x-1)^3} = \underbrace{x + 2a + 3}_{q(x)} + \dfrac{(a^2+6a+2b+6)x^2 + (2ab-6a-8)x +2a+b^2+3}{(x-1)^3}

If the remainder is supposed to be zero, then

\begin{cases}a^2+6a+2b+6 = 0 \\ 2ab-6a-8 = 0 \\ 2a+b^2+3 = 0\end{cases}

Adding these equations together and grouping terms, we get

(a^2+2ab+b^2) + (2a+2b) + (6-8+3) = 0 \\\\ (a+b)^2 + 2(a+b) + 1 = 0 \\\\ (a+b+1)^2 = 0 \implies a+b = -1

Then b=-1-a, and you can solve for <em>a</em> and <em>b</em> by substituting this into any of the three equations above. For instance,

2a+(-1-a)^2 + 3 = 0 \\\\ a^2 + 4a + 4 = 0 \\\\ (a+2)^2 = 0 \implies a=-2 \implies b=1

So, we end up with

p(x) = x^2 - 2x + 1 \\\\ \implies f(x) = x^3 (x^2 - 2x + 1) = \boxed{x^5-2x^4+x^3}

You might be interested in
Y = -6x + 3 y = 5x - 8 Question 1 options: (-3, -1) (3, -1) (1, -3) (1, 3)
OleMash [197]

Answer:

-3,-1 so a

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
A sphere and a cylinder have the same radius and height. The volume of the cylinder is 18 cm.
Oksi-84 [34.3K]

Answer:

38cm3

Step-by-step explanation:

8 0
3 years ago
Why is the relationship between the u.s and foreign nations important?
Leni [432]

Foreign relations are important because they are meant to prevent misunderstandings and future conflicts between nations. They strengthen the economy and the common goals between countries, becoming the root of great binding agreements. Foreign relations plays a key part in order to achieve peace one day.

3 0
3 years ago
371.616 what is the number in the ones placed
Elina [12.6K]
The number in the ones place is 1.
7 0
3 years ago
Read 2 more answers
5. Find a counterexample to show that the following con-
Bess [88]

Answer:  see below

<u>Step-by-step explanation:</u>

Any value x such that |x| < 1 will make the conjecture false

<em>In simpler words, let x be a fraction between 0 and 1.</em>

One example: Let x = \dfrac{1}{2}

Then x⁴ = \bigg(\dfrac{1}{2}\bigg)^4

             = \dfrac{1}{16}

\dfrac{1}{16} < \dfrac{1}{2}

so the conjecture is false.

7 0
3 years ago
Other questions:
  • Jeremiah received $65 for his birthday and used part of it to buy 3 video games. The games cost the same amount, and he had $14
    15·2 answers
  • Solve the equation for x. If a solution is extraneous, be sure to identify it in your final answer.
    13·1 answer
  • WILL GIVE BRAINLIEST AND 40 POINTS! NEED THIS ASAP!
    6·2 answers
  • Suppose you roll a regular 6 faced die what is the probability of getting a 2,6,4
    7·1 answer
  • 7/20 in decimal form
    7·2 answers
  • Which matrix equation has the solution
    15·1 answer
  • Which is the slope of the line that passes through the points (−3,1) and (4,−2)?
    5·2 answers
  • Workers who drive to work in a city have a one in three chance of experiencing traffic delays.
    10·2 answers
  • Complete the proof by providing the missing statement and reasons
    10·1 answer
  • Which equations support the fact that rational numbers are closed under subtraction?
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!