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marta [7]
3 years ago
13

The polynomial $f(x)$ has degree 3. If $f(-1) = 15$, $f(0)= 0$, $f(1) = -5$, and $f(2) = 12$, then what are the $x$-intercepts o

f the graph of $f$?

Mathematics
1 answer:
Alex_Xolod [135]3 years ago
4 0

Your calculator's cubic regression function can tell you the equation is

... f(x) = 2x³ + 5x² -12x = x(x +4)(2x-3)

The x-intercepts are -4, 0, +1.5.

_____

If you want to solve this "by hand", you can first of all recognize that since there is an x-intercept at 0, the cubic will only have three coefficients. That is, you can write the equation as

... f(x) = ax³ + bx² + cx

Substituting the given points (except (0, 0)) gives three linear equations in a, b, c.

... -a +b -c = 15 . . . . . for x=-1

... a + b + c = -5 . . . . for x=1

... 8a +4b +2c = 12 . . for x=2

adding the first two equations gives 2b=10, or b=5. Now, you can reduce the system to

... a + c = -10

... 4a +c = -4

Subtracting the first of these equations from the second gives 3a=6, or a=2. That tells you c=-12 (from a+c=10).

Then your equation is

... f(x) = x(2x² +5x -12)

Factoring by any of the usual techniques, or graphing, or using the quadratic formula will tell you the zeros (x-intercepts) are as above.

_____

Since the input values are sequential, you can also develop the function from differences of the output values. Those are 15, 0, -5, 12. First differences are -15, -5, +17. Second differences are +10, +22. The third difference is 12. Using the first of these differences in appropriate places in the interpolating polynomial formula, we have

... f(x) = 15 + (x+1)(-15 + (x)/2·(10 + (x-1)/3·(12))) = 2x³ +5x² -12x . . . . as above

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Factor by grouping.......
Gekata [30.6K]

Answer:

A. (7n-1) (8n-1)

Step-by-step explanation:

56n^2 -8n-7n+1

56n^2 -8n       -7n+1

Factor out 8n from the first group   and -1 from the second group

8n (7n-1)     -1(7n-1)

Factor out the (7n-1)

(7n-1) (8n-1)

7 0
3 years ago
What is the 38th term of 459,450,441,..
galina1969 [7]

Answer:

The 38th term of 459,450,441,.. will be:

a_{13}=351

Step-by-step explanation:

Given the sequence

459,450,441,..

An arithmetic sequence has a constant difference 'd' and is defined by

a_n=a_1+\left(n-1\right)d

computing the differences of all the adjacent terms

450-459=-9,\:\quad \:441-450=-9

so

d=-9

The first element of the sequence is

a_1=459

so the nth term will be

a_n=-9\left(n-1\right)+459

a_n=-9n+468

Putting n=38 to find the 38th term

a_n=-9n+468

a_{13}=-9\left(13\right)+468

a_{13}=-117+468

a_{13}=351

Therefore, the 38th term of 459,450,441,.. will be:

a_{13}=351

3 0
3 years ago
The number of accidents that occur at a busy intersection is Poisson distributed with a mean of 4.5 per week. Find the probabili
Mariana [72]

Answer:

a) 0.01111

b) 0.4679

c) 0.33747

Step-by-step explanation:

We are given the following in the question:

The number of accidents per week can be treated as a Poisson distribution.

Mean number of accidents per week = 4.5

\lambda= 4.5

Formula:

P(X =k) = \displaystyle\frac{\lambda^k e^{-\lambda}}{k!}\\\\ \lambda \text{ is the mean of the distribution}

a) No accidents occur in one week.

P(x =0)\\\\= \displaystyle\frac{4.5^0 e^{-4.5}}{0!}= 0.01111

b) 5 or more accidents occur in a week.

P( x \geq 5) = 1-\displaystyle \sum P(x

c) One accident occurs today.

The mean number of accidents per day is given by

\lambda = \dfrac{4.5}{7} = 0.64

P(x =1)\\\\= \displaystyle\frac{0.64^1 e^{-0.64}}{1!}= 0.33747

5 0
3 years ago
Given: ∠JOH = 27° and arc NML = 98°. Find the measure of arc JH.
stepladder [879]
We need an image because of this

6 0
3 years ago
Find the hypotenuse of a 45-45-90 triangle with legs equal to 7
madam [21]

Answer:

Square root of 98

Step-by-step explanation:

5 0
3 years ago
Read 2 more answers
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