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Featured 3 months ago

According to Factor theorem, for a polynomial

Hence, if we have a polynomial say

Observe that coefficient of highest power of

In practice, Factor Theorem is used for factorising polynomials completely. So our steps in factorising a polynomial

1 Check the constant term in

2. Find its all possible factors.

3. Take one of the factors, say

4. Try more factors whose number should be equal to the degree of polynomial.

And you have got all the factors.

Coming to your question, we have

or

or

This is the same as in

We should have zeros among

Note that it is easier to check whether

Featured 3 months ago

One should complete the squares so that the equation may be written in one of the two following forms:

where

Given:

Add 40 to both sides:

Group the x terms and the y terms together:

We cannot complete the square unless the leading coefficient is 1, therefore, we remover a factor of 8 from the y terms:

Because

Matching the x terms with the general pattern,

will allow us to solve for the value of h:

This means that

Combine like terms:

We want insert

Matching the y terms with the general pattern,

will allow us to solve for the value of k:

This means that

Combine like terms:

Divide both sides by 64:

Write the denominators as squares:

This is the same form as equation [1],

The center,

The vertices are,

The foci are,

The eccentricity is

Featured 2 months ago

Determination of the zeros and a sign chart of

**Finding the zeros of #f(x)=x^3(x+2)^2#**

Remember that a function has the value zero at (and only at) points where some factor of the function has the value zero.

The factors of

are

and the only unique factors are

Therefore

For non-zero values of

We can pick arbitrary values within each interval to determine if

**Unfortunately this does **not** tell us any detailed information about the behavior of

Just for reference, here is what the graph should look like, but you would need to use something beyond the zeros and a sign chart to sketch this.

Featured 1 month ago

Recall that, for the **Hyperbola**

asymptotes are given by,

In our **Case,** the asymptotes are,

Next,

Multiplying by

By

With **Hyperbola** is,

Featured 1 month ago

**Solution set**

Alternate notation:

**Solve:**

**1) Examine the related equation to find the boundaries.**

or

The solutions to the related equation define boundary points that divide the number line into 5 parts, 3 regions and the 2 boundary points.

- Since the inequality is inclusive ("non-strict") the boundary points are in the solution set.

A graph of the number line, with solid dots on (-1), and (5) will help in thinking about the problem.

To check for the solution, **use test points, one from each region, in the inequality to see if they qualify:**

Try (-2), (0), and (6). It is best not to pick numbers that make the arithmetic difficult.

**In region,**

**In region,**

**In region,**

We must be careful in stating the solution inequalities. x can not be in both regions at once, since they are disconnected. There is a full region between the two regions in the solution, so we can't use "and", **we must use "or"**.

**Solution set**

Alternate notation:

**2-D Graph of the inequality:**

Featured 1 month ago

Perform the Gauss Jordan elimination on the augmented matrix

I have written the equations not in the sequence as in the question in order to get

Perform the folowing operations on the rows of the matrix

Thus

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