salt water has a density of 1030 kg/m^3. a piece of wood with mass of 1260 g is a cuboid with dimensions 7 cm x 10 cm x 15 cm.Wi
1 answer:
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<h3>Answer:</h3>
- zeros are {-2, 3, (-1±i√3)/2}
I like to look at a graph of the function to see where the zeros might be. Here, there are x-intercepts at x=-2 and x=3. These can be factored out using synthetic division to find the factorization to be ...
... (x +2)(x -3)(x² +x +1) = 0
By completing the square, using the quadratic formula, or by looking at the graph of it, the complex roots of the quadratic factor can be found to be ...
... x = (-1 ±i√3)/2
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The second attachment shows my synthetic division. The first division takes out the root x=3 to give a quotient of x³ +3x² +3x +2. The second division takes out the root -2 to give the quotient of x² +x +1. (You can see that I tried -1 as a root first.)
The graph shows both the quartic and the quadratic factor of it. The latter has a leading coefficient of 1 and a vertex at (-1/2, 3/4), so you know the complex roots are -1/2 ±i√(3/4).
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<em>From the beginning</em>
There is only a very complicated formula for the roots of a quartic equation, so these are usually solved by machine or by some form of trial and error (iteration). There are some helps, like Descarte's Rule of Signs, and the Rational Root theorem.
Here, the former looks at the one sign change in the coefficients to tell you there will be 1 positive real root. Changing the sign of the odd-degree terms makes there be 3 sign changes, so there will be 3 or 1 negative real roots. Thus, we're assured at least two real roots, one of each sign.
We can look at the constant term to find the y-intercept to be -6. We can add the coefficients to find the value of the function is -18 for x=1, so the positive real root is larger than 1.
The Rational Root theorem says any rational roots will be factors of 6, the constant term. Choices are 1, 2, 3, 6. We have already eliminated 1 as a possibility, and we consider it unlikely that 6 will be a root. (The 4th power overwhelms the other terms in the function.) We tried 2 and found it doesn't work (this was before we graphed the function). The attached division result shows that 3 is a root, as does the graph.
Once you get down to a quadratic, you can find the remaining roots in the usual way. Because it is so simple to read them from the graph, we decided to graph the quadratic factor.
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<em>Comment on terminology</em>
"root" and "zero" are essentially the same thing when the function is equated to zero, as here. The terms refer to the value(s) of x that make the polynomial function evaluate to zero.
Answer:
Option (B)
Step-by-step explanation:
Given sequence is 45, 135, 405, 1215, 3645..........
Since it's a geometric sequence,
Common ratio of each successive term to the previous term
r =
First term of the sequence 'a' = 45
Explicit formula of a geometric sequence will be,
Recursive formula of this sequence will be,
,
Therefore, Option (B) will be the answer.
Answer:
The answer: The factors are (2x-9) and (x+6).
The Problem:
Factor
Step-by-step explanation:
So I'm going to do trial factors using the choices to aid me.
Factored form for this problem if it exist will be in the form:
.
In general this is what it would look like if we factored any quadratic in terms of (given the quadratic is not prime but technically you could factor even over the complex numbers).
Let's look at:
We want to choose such that when you multiply them you get 2. Well those would have to be 2 and 1.
we want to choose such that when you multiply them you get -54. Based on the choices we want to get with -9 and 6, or 9 and -6. We don't know the order we want to choose it in either.
For example which of these would work:
We are going to consider only the outer and inner of FOIL since we already know the first times the first is and the last times the last is
.
Let's test the first one:
Outer: 2x(9)=18x
Inner: -6(x)=-6x
------------------------ADD!
18x-6x=12x
The first choice did not give us the middle term 3x.
Trying the second one would give us the opposite since they are in the same form as previous just the + and - are switched.
Let's look at the third one:
Outer: 2x(6)=12x
Inner: -9(x)=-9x
--------------------------ADD!
12x-9x=3x
This is the winner.
The answer: The factors are (2x-9) and (x+6).