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adell [148]
3 years ago
10

What is the least possible degree of a polynomial that has roots -5,1 + 4i, and -4i?

Mathematics
1 answer:
Oksanka [162]3 years ago
6 0

Without any extra conditions, the answer could be 3, and the simplest polynomial with the given roots would be

(<em>x</em> + 5) (<em>x</em> - (1 + 4<em>i</em> )) (<em>x</em> + 4<em>i</em> )

= <em>x</em> ³ + 4<em>x</em> ² + (11 - 4<em>i</em> ) <em>x</em> + 80 - 2<em>i</em>

<em />

If the polynomial is supposed to have only <em>real</em> coefficients, then any complex roots must occur along with their complex conjugates:

(<em>x</em> + 5) (<em>x</em> - (1 + 4<em>i</em> )) (<em>x</em> - (1 - 4<em>i</em> )) (<em>x</em> + 4<em>i</em> ) (<em>x</em> - 4<em>i </em>)

= <em>x</em> ⁵ + 3<em>x</em> ⁴ + 23<em>x</em> ³ + 133<em>x</em> ² + 112<em>x</em> + 1360

and then the degree would be 5.

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AM is a median in △ABC (M∈ BC ). A line drawn through point M intersects AB at its midpoint P. Find areas of △APC and △PMC, if A
Snowcat [4.5K]

Answer:

The area of APC is 70m². The area of triangle PMC is 35m².

Step-by-step explanation:

Let the area of triangle ABC be x.

It is given that AM is median, it means AM divides the area of triangle in two equal parts.

\text{Area of }\triangle ACM=\text{Area of }\triangle ABM=\frac{x}{2}    .....(1)

The point P is the midpoint of AB, therefore the area of APC and BPC are equal.

\text{Area of }\triangle APC=\text{Area of }\triangle BPC=\frac{x}{2}          ......(2)

The point P is midpoint of AB therefore the line PM divide the area of triangle ABM in two equal parts. The area of triangle APM and BPM are equal.

\text{Area of }\triangle APM=\text{Area of }\triangle BPM=\frac{x}{4}        .....(3)

The area of triangle APM is 35m².

\text{Area of }\triangle APM=\frac{x}{4}

35=\frac{x}{4}

x=140

Therefore the area of triangle ABC is 140m².

Using equation (2).

\text{Area of }\triangle APC=\frac{x}{2}

\text{Area of }\triangle APC=\frac{140}{2}

\text{Area of }\triangle APC=70

Therefore the area of triangle APC is 70m².

Using equation (3), we can say that the area of triangle BPM is 35m² and by using equation (2), we can say that the area of triangle BPC is 70m².

\triangle BPC=\triangle BPM+\triangle PMC

70=35+\triangle PMC

35=\triangle PMC

Therefore the area of triangle PMC is 35m².

8 0
3 years ago
What is the slope of the line that passes through the points (-10,8)and (-15,-7)
balandron [24]

Answer:

m = 3

Step-by-step explanation:

Use the slope formula to find the slope  m .

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Answer:

16

Step-by-step explanation:

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4 x 4 = 16

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How long will it take a seal swimming at a speed of 8 miles per hour to travel 52miles
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Let $f(x)$ be the real-valued function defined for all real $x$ except for $x = 0$ and $x = 1$ and satisfying the functional equ
Mademuasel [1]
We have to find the values of F.
In this case. F is unlikely to be a polynomial.
But the problem is, we can’t calculate the values of F directly.
There is no real value of x for which x = x−1 x because F isn’t defined at 0 or 1. so,
substituting x = 2.
F(2) + F(1/2) = 3.

Substitute, x = 1/2
F(1/2) + F(−1) = −1/2.
We still are not getting the required value,
therefore,
Substitute x = −1

As, F(2) +F(−1) = 0.
now we have three equations in three unknowns, which we can solve.
It turns out that:
F(2) = 3/4
F(3) = 17/12
F(4) = 47/24
and
F(5) = 99/40

Setting
g(x) = 1 − 1/x
and using
2 → 1/2
to denote
g(2) = 1/2
 we see that :
x → 1 - 1/x → 1/(1-x) →x

so that:
g(g(g(x))) = x.

Therefore, whatever x 6= 0, 1 we start with, we will always get three equations in the three “unknowns” F(x), F(g(x)) and F(g(g(x))).
Now solve these equations to get a formula for F(x)

As,
h(x) = (1+x)/(1−x)
which satisfies h(h(h(h(x)))) = x

Now, mapping x to h(x) corresponds to rotating the circle by ninety degrees.

7 0
3 years ago
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