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Kay [80]
3 years ago
12

How many distinct and real roots can an nth-degree polynomial have?

Mathematics
1 answer:
Vlad [161]3 years ago
8 0
The fundamental theorem of algegra is often cited to say that an n-th degree polynomial has n roots. They may not always be distinct, and they may not always be real.

An n-th degree polynomial may have up to n distinct real roots.
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Is the dilation from A to B a reduction or enlargement?
WITCHER [35]

Answer:

This is actually a reduction, and the scale factor of this dilation is 0.5.

8 0
2 years ago
Read 2 more answers
Find the integral using substitution or a formula.
Nadusha1986 [10]
\rm \int \dfrac{x^2+7}{x^2+2x+5}~dx

Derivative of the denominator:
\rm (x^2+2x+5)'=2x+2

Hmm our numerator is 2x+7. Ok this let's us know that a simple u-substitution is NOT going to work. But let's apply some clever Algebra to the numerator splitting it up into two separate fractions. Split the +7 into +2 and +5.

\rm \int \dfrac{x^2+2+5}{x^2+2x+5}~dx

and then split the fraction,

\rm \int \dfrac{x^2+2}{x^2+2x+5}~dx+\int\dfrac{5}{x^2+2x+5}~dx

Based on our previous test, we know that a simple substitution will work for the first integral: \rm \quad u=x^2+2x+5\qquad\to\qquad du=2x+2~dx

So the first integral changes,

\rm \int \dfrac{1}{u}~du+\int\dfrac{5}{x^2+2x+5}~dx

integrating to a log,

\rm ln|x^2+2x+5|+\int\dfrac{5}{x^2+2x+5}~dx

Other one is a little tricky. We'll need to complete the square on the denominator. After that it will look very similar to our arctangent integral so perhaps we can just match it up to the identity.

\rm x^2+2x+5=(x^2+2x+1)+4=(x+1)^2+2^2

So we have this going on,

\rm ln|x^2+2x+5|+\int\dfrac{5}{(x+1)^2+2^2}~dx

Let's factor the 5 out of the intergral,
and the 4 from the denominator,

\rm ln|x^2+2x+5|+\frac54\int\dfrac{1}{\frac{(x+1)^2}{2^2}+1}~dx

Bringing all that stuff together as a single square,

\rm ln|x^2+2x+5|+\frac54\int\dfrac{1}{\left(\dfrac{x+1}{2}\right)^2+1}~dx

Making the substitution: \rm \quad u=\dfrac{x+1}{2}\qquad\to\qquad 2du=dx

giving us,

\rm ln|x^2+2x+5|+\frac54\int\dfrac{1}{\left(u\right)^2+1}~2du

simplying a lil bit,

\rm ln|x^2+2x+5|+\frac52\int\dfrac{1}{u^2+1}~du

and hopefully from this point you recognize your arctangent integral,

\rm ln|x^2+2x+5|+\frac52arctan(u)

undo your substitution as a final step,
and include a constant of integration,

\rm ln|x^2+2x+5|+\frac52arctan\left(\frac{x+1}{2}\right)+c

Hope that helps!
Lemme know if any steps were too confusing.

8 0
3 years ago
A factory is going to hire 33 new employees. There are 79 applicants. Which
dedylja [7]

Answer:

33:79

Step-by-step explanation:

We are told that a factory is going to hire 33 new employees.

and the total applicants is 79

In this problem, we want to represent the situation as a ratio of new hires to applicants

Since the question says  new hires to applicants

Hence, the ratio will be

33 to 79

or

33/79, we can simplify this as

33:79

5 0
2 years ago
In order to find constant rate, you need to use change in y ÷ change in x right?
NemiM [27]

Yes u have to and as long are you do than you would be right

3 0
3 years ago
Given the two Fibonacci numbers below, which number would follow? F(14) = 377 and F(15) = 610
beks73 [17]

Answer:

987.

Step-by-step explanation:

The terms in a Fibonacci series are obtained form the sum of the previous 2 terms. So the next number F(16) = 377 + 610 = 987.

6 0
3 years ago
Read 2 more answers
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