Given:
The polynomials are:
To find:
The completely simplified sum of the polynomials.
Solution:
We have,
The sum of given polynomials is:
Therefore, the sum of the given polynomials is . It is a polynomial with degree 6 and leading coefficient -2.
If Kate has an apple, an orange, a pear, a banana, and a kiwi at home and she wants to bring three pieces of fruit to school, ho
2 answers:
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A good place to start is to set to y. That would mean we are looking for
to be an integer. Clearly,
, because if y were greater the part under the radical would be a negative, making the radical an imaginary number, not an integer. Also note that since
is a radical, it only outputs values from
, which means y is on the closed interval:
.
With that, we don't really have to consider y anymore, since we know the interval that is on.
Now, we don't even have to find the x values. Note that only 11 perfect squares lie on the interval , which means there are at most 11 numbers that x can be which make the radical an integer. All of the perfect squares are easily constructed. We can say that if k is an arbitrary integer between 0 and 11 then:
Which is strictly positive so we know for sure that all 11 numbers on the closed interval will yield a valid x that makes the radical an integer.