Previously, we learned patterns for squaring and cubing binomials. (x+y) 3 = x 3 + 3x 2 y + 3xy 2 + y 3. (x+y) 2 = x 2 + 2xy + y 2. If the patterns were not memorized, you could simply write it out and multiply. (x+y) 2 = (x + y)(x + y). (x+y) 3 = (x + y)(x + y)(x + y).
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Previously, we learned patterns for squaring and cubing binomials.
(x+y)3= x3 + 3x2y + 3xy2 + y3
(x+y)2= x2 + 2xy + y2
If the patterns were not memorized, you could simply write it out and multiply.
(x+y)2= (x + y)(x + y)
(x+y)3= (x + y)(x + y)(x + y)
The square of a binomial is not a problem, just foil. The cube of a binomial is a little work. But any exponent higher than 3 is a problem. That would just be too much work to multiply 4 binomials.
(x+y)4= (x + y)(x + y)(x + y)(x + y) Too much work! Agreed?
is read n choose r, “combinations of n things taken r at a time.”
For example if we had 6 people, how many different groups of 2 could we choose?
We can calculate this on a scientific calculator. We will show how to perform this operation on a TI-30XIIS and a graphing calculator. If you calculator different, please go to an assistant or your instructor.
Fortunately we have an alternative method which is the Binomial Theorem. Before we learn the binomial theorem, we need to have a little fun playing with our scientific calculators. We will only show what you need to “get by” on this section. We will not get involved with factorials, permutations, and combinations. We would give more detail on these topics in a course covering probability and statistics.
1st the TI-30XIIS.
2nd, a graphing calculator.
Things to note:
1) The powers of x are decreasing while the powers of y are increasing.
2) For each term, the sum of the exponents on the x and y expressions is n.
3) The top number in the binomial coefficient always equals n.
4) For each term, the exponent on the y expression and bottom number in the binomial
coefficient are always one less than the number of the term.
The Binomial Theorem
For algebraic expressions, x and y, and any natural number, n,
n = 6
Your Turn Problem #1
Expand (x + y)8
Example 1: Use the binomial theorem to multiply (expand): (x + y)6
Note: the x and y may not be variables. They can be any algebraic expression.
Example 2. Use the binomial theorem to multiply (expand): (x + 7)5
; (substitute 7 for y in the binomial expansion)
(x + 7)5
Your Turn Problem #2
Expand (x + 5)7
Note: the x and y can represent terms with variable powers.
Example 3. Use the binomial theorem to multiply (expand): (5a2 + 4b3)5
; (substitute 5a2 for x and 4b3 for y in the binomial expansion)
Your Turn Problem #3
Expand (9m5 + 3k2)4
Note: the terms can have negative signs.
Example 4. Use the binomial theorem to multiply (expand): (2p3 – 5)7
(note: since one line of these problems is a little long, you may want to turn your paper sideways.)
Your Turn Problem #4
Expand (4n2 – 7)6
(2p3 – 5)7 ; (substitute 2p3 for x and -5 for y in the binomial expansion)