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-Dominant- [34]
3 years ago
10

(-2.5,-4)and(1.5,-2)

Mathematics
1 answer:
svlad2 [7]3 years ago
3 0

Slope-

m = 0.5

--

Midpoint-

(-0.5, -3)

--

Distance between two points-

4.47213595

--

Equation of the line-

y = 0.5x - 2.75

--

y intercept-

b = -2.75

--

Graph-


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Find a polynomial $f(x)$ of degree $5$ such that both of these properties hold: $\bullet$ $f(x)$ is divisible by $x^3$. $\bullet
frosja888 [35]

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}

6 0
2 years ago
What is expanded form of 605,970
Luba_88 [7]
605,970 is expanded in this form:

(6 x 100,000) + (0 x 10,000) + (5 x 1,000) + (9 x 100) + (7 x 10) + (0 x1)

each number is multiplied to the corresponding place value.
6 is on the one hundred thousand place value
0 is on the ten thousand place value
5 is on the one thousand place value
9 is on the hundreds place value
7 is on the tens place value
0 is on the ones place value

OR

600,000 + 5,000 + 900 + 70 = 605,970
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A delivery truck weighs 5.5 tons before it is loaded with packages. Once the delivery truck is full of x packages it weighs 6.1
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Answer:

80 packages

Step-by-step explanation:

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Which expression has the same value as 15+24
lakkis [162]
15+24= put it in the calculator u will find the answer(hope this helped)
6 0
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Find the common difference of the arithmetic sequence 13, 16, 19,...13,16,19,...
VLD [36.1K]

Answer:

the common difference is plus 3

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