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Ad libitum [116K]
3 years ago
7

If f(1) = 0, what are all the roots of the function f(x)=x^3+3x^2-x-3? Use the Remainder Theorem.

Mathematics
2 answers:
Sophie [7]3 years ago
8 0
There's no if about it, 

f(x)=x^3+3x^2-x-3


has a zero f(1)=0 so x-1 is a factor.   That's the special case of the Remainder Theorem; since f(1)=0 we'll get a remainder of zero when we divide f(x) by x-1.

At this point we can just divide or we can try more little numbers in the function.  It doesn't take too long to discover f(-1)=0 too, so  x+1 is a factor too by the remainder theorem.  I can find the third zero as well; but let's say that's out of range for most folks.

So far we have 

x^3+3x^2-x-3 = (x-1)(x+1)(x-r)

where r is the zero we haven't guessed yet.  Again we could divide f(x) by (x-1)(x+1)=x^2-1 but just looking at the constant term we must have

-3 = -1 (1)(-r) = r

so

x^3+3x^2-x-3 = (x-1)(x+1)(x+3)

We check f(-3)=(-3)^3+3(-3)^2 -(-3)-3 = 0 \quad\checkmark

We usually talk about the zeros of a function and the roots of an equation; here we have a function f(x) whose zeros are

x=1, x=-1, x=-3

max2010maxim [7]3 years ago
7 0

I think you got it backwards! Its <em>X=-3, X=-1, or X=1.</em> So the answer is B. (2nd option) Well, you got it! ;)

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Alex

Answer:

Let us say the domain in the first case, has the numbers. And the co-domain has the students, .

Now for a relation to be a function, the input should have exactly one output, which is true in this case because each number is associated (picked up by) with only one student.

The second condition is that no element in the domain should be left without an output. This is taken care by the equal number of students and the cards. 25 cards and 25 students. And they pick exactly one card. So all the cards get picked.

Note that this function is one-one and onto in the sense that each input has different outputs and no element in the co domain is left without an image in the domain. Since this is an one-one onto function inverse should exist. If the inverse exists, then the domain and co domain can be interchanged. i.e., Students become the domain and the cards co-domain, exactly like Mario claimed. So, both are correct!

8 0
3 years ago
Find the derivative of the function. y = sin−1(6x + 1)
liraira [26]

If you don't know the derivative of the inverse of sine, you can use implicit differentiation. Apply sine to both sides:

y=\sin^{-1}(6x+1)\implies\sin y=6x+1

(true for <em>y</em> between -π/2 and π/2)

Now take the derivative of both sides and solve for it:

\cos y\dfrac{\mathrm dy}{\mathrm dx}=6

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7 0
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Adam is training a group of cyclists. Each cyclist completes a 10-kilometer time trial during training. Adam records the times,
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The histogram which shows the results of the time trial is option B

<h3>Histogram</h3>

Given data:

12:32, 12:46, 13:30, 15:03, 13:43, 13:46, 14:20, 14:29, 12:34, 14:43, 15:26, 16:02, 16:16, 14:38, 16:55, 17:10, 14:26, 17:21, 17:58, 13:30

Grouping of data,:

12:00 - 12:59 = 3

13:00 - 13:59 = 4

14:00 - 14:59 = 5

15:00 - 15:59 = 2

16:00 - 16:59 = 3

17:00 - 17:59 = 3

  • The frequency is represented on the y-axis(number of cyclist)
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Learn more about histogram:

brainly.com/question/9388601

#SPJ1

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Answer:

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Step-by-step explanation:

i did the assignment and got it correct

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