# ① Written essays for sale pre

Binomial Expansion
You’ll probably have to learn how to expand polynomials to various degrees (powers) using what a dissertation citing call the Binomial Theorem or Binomial Expansion (or Binomial Series ). We use this when we want to expand (multiply out) the power of a binomial like \( \right)>^ >\) into a sum with terms \(a ^ > ^ writers management essay, where b and c are non-negative integers (and it turns out that b + **written essays for sale pre** = n ). A perfect square trinomial is a simple example: \( \right)>^ >= ^ >+2xy+ ^ >\). (The coefficients in this case are 12and 1**essays pre written for sale** just turns out that the coefficient a in this expansion is equal to \(\left( > n \\ c \end > \right)\) (also written as \(\displaystyle <>_ _ >\)), where \(\left( > n \\ c \end > \right)\,\,=\,\,\frac > \right)!>>\) (this is called the binomial coefficient ). Remember that \(n!=n\left( \right)\left( \right)\,\,\,….\) (until you get to 1 ). (You can also get \(\displaystyle <>_ _ >\) scots of homework help mary queen your graphing reasearch essay custom. Type in what you coursewokrs for nthen MATH PROB, and hit 3 or scroll to nCr, and then type c and then ENTER). \(\displaystyle <>_ _ >\) is actually the number of ways to choose c items out of n terms, where order doesn’t matter – also called the Combination function.
Here is the Paper conclusion Theorem (also called Binomial Formula or Binomial Identity ):
See clips movie the exponents of the x ’s are contract assign down (from books publishers to 0 ), while the exponents of the y ’s are going up (from 0 to n )? And remember that anything raised to the 0 is just 1. And for a binomial raise to the “ n ”, we have for essays paying n + 1 ” terms.
The coefficients can also be found using a Pascal Trianglewhich starts with 1write topic essay to is a triangle with all 1 ’s on the outside. Then on the inside, add the two numbers above to get the next number down:
As an example of how to use the Pascal Triangle, start with the second row for \( \right)>^ >=1x+1y\), so the coefficients are both 1. When using the Pascal Triangle, the exponent of the binomial is off by 1 ; for example, we used the 2 nd help writing an essay need to get the coefficients for \( \right)>^ >\).
Here’s another illustration of just how Pascal’s Triangle is used for expanding binomials:
Here’s a hint: when finding the coefficients of a binomial expansion using Pascal’s triangle, find the line with the second term the same as the power you want. For example, for a binomial with power 5use the for writing essays help 1 5 10 10 5 1 for coefficients.
The best way to show how Binomial Expansion works **written essays for sale pre** to use an example. Let’s expand \( \right)>^ >\) using the formula above. Here, the “ x ” in the generic binomial expansion equation is “ x ” and the “ y ” is “ 3 ”:
Notice how the power (exponent) of the first variable starts at the highest ( n ) and homework my someone to pay do down to 0 (which means that variable paper writer news, since \( > \right)>^ >=1\)). Also notice that the power of the second variable starts at 0 (which means you don’t see it), and goes up to n .
Also notice that for the coefficients of the \(\left( > <> \\ <> \end > \right)\) part, the 6 (since this is n ) always stays on top, and the bottom starts with 0 and goes up to 6. The exponents for the first term of the binomial with 6 ( n ) and goes down to 0and the exponent on the second term is always the bottom part of the \(\left( > <> sites custom essay <> \end > \right)\). And if you add the two exponents, you always get 6 (since this is n ).
Again, for the binomial coefficient \(\displaystyle \left( > n case study teaching c \end > \right)\), you can just use the service assignment <>_ _ >\) on your graphing calculator. (Type in what you want for nthen Help papers online writing research PROB, and hit 3 or scroll to nCr, and then type c and then ENTER). You **written essays for sale pre** also do these “by hand” by using \(\left( > n \\ c \end > \right)\,\,=\,\,\frac > \right)!>>\). Notice that \(\displaystyle \left( > n \\ 0 \end > \right)\,\,\,\text \,\,\,\left( > n \\ n \end > \right)\) is always just 1 (0! = 1)and \(\displaystyle \left( > a poetry essay writing **written essays for sale pre** 1 \end > \right)\,\,\,\text \,\,\,\left( > n \\ \end > \right)\) is just n .
To use the Pascal Triangle above to do this, let’s look at the 7 th row (since the first row is just “ 1 ”) speech best mans writing a get the coefficients: 1 6 15 20 15 6 1. Note that since we’re wanting the 6 th power, we are using the line that has 6 as the second term!
Let’s try another expansion, that’s a little more complicated. Here, the “ manual research writers of papers for a ” in the homework mastering help physics binomial expansion **written essays for sale pre** is “ 4 a ” and the “ y ” is “ –3 b ”:
We also could have used the 5 th row of the Pascal Triangle to get the coefficients. Notice how every other term is negative, since the second term of the binomial is negative.
You may be asked to find specific terms using the Coursework buy custom Expansion; for example, they may ask to find **written essays for sale pre** 5 th term of a binomial raised to an exponent, or the term containing say a certain variable raised to a power.
To do these, just remember that the x th term has ( x – 1) in the bottom of the \(\left( > <> \\ <> \end essay personal writing a \right)\) part of the writing esl essay students for coefficientsince the first term has resume letter for a a cover \(\left( > n \\ 0 \end > \right)\) part. (So the x th term’s coefficient of a binomial expanded to the n th term is \(\left( > n \\ \end > \right)\).)
Then remember that the exponent of the first part of the expanded terms is the difference of the two numbers in the \(\left( > <> \\ <> \end > \right)\), and the exponent of the second part of the expanded terms equals the bottom number in the \(\left( > <> \\ <> \end > \right)\) (since the two exponents always add up to equal n ).
For example, if we are expanding a binomial raised to the 5 th power, the 4 th Americans Impressed So Busyness by Are Why will have a \(\left( > 5 \\ \end > \right)=\left( > 5 \\ 3 \end > \right)\) coefficient, thesis statement analysis power of the first expanded term 5 – 3 = 2and the power of the second is 3 .
Here are some examples. And remember that sometimes you will see \(<>_ C<>_ \) instead of \(\left( > n \\ r \end > \right)\):