Example Questions

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Example Question #11 : Non Quadratic Polynomials

Factor

Explanation:

First, we can factor a  from both terms:

Now we can make a clever substitution.  If we make  the function now looks like:

This makes it much easier to see how we can factor (difference of squares):

The last thing we need to do is substitute  back in for , but we first need to solve for  by taking the square root of each side of our substitution:

Substituting back in gives us a result of:

Example Question #171 : Polynomials

Simplify:

Explanation:

Use the difference of squares to factor out the numerator.

The term  is prime, but  can still be factored by another difference of squares.

Replace the fraction.

Simplify the top and bottom.

Example Question #171 : Polynomials

Find a polynomial function  of the lowest order possible such that two of the roots of the function are:

Explanation:

Find a polynomial function  of the lowest order possible such that two of the roots of the function are:

Recall that by roots of a polynomial we are referring to values of  such that

Because one of the roots given is a complex number, we know there must be a second root that is the complex conjugate of the given root. This is

Because  is a root, the unknown function  must have a factor

The other roots are complex numbers, so there must be a quadratic factor.

Isolate the imaginary term  onto one side and square,

Expand the left side, and note on the right side the  factor reduces as follows:

So now we have,

The quadratic factor for  is therefore  . Combining this with the  factor give a factored expression for the desired function:

Now we carry out the multiplication to write the final form of

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