Unit 2 - 5 Binomial Expansion
What is the general formula for binomial function?
(a+b)^(n) = a^(n) + (n/1)a^(n−1)b + (n/2)a^(n−2)b^(2) + ...+(n/r)a^(n−r)b(r) +...+bn Where n∈ℕ
How can nCr be written?
(n/r) but like a vector
What equals 0!?
1
What is always the coefficient of the first and last term in the bionmial series?
1
What is the first row of pascals triangle? n=0
1
What is the second row of pascals triangle? n=1
1 , 1
What is the third row of pascals triangle? n=2
1, 2, 1
What is the fourth row of pascals triangle? n=3
1, 3, 3, 1
What is the fifth row of pascals triangle? n=4
1, 4, 6, 4, 1
What happens to the powers of b in binomial theorem for (a+b)^n?
The power of b starts at 0 and increases by 1 for each term, so the powers are 0, 1, 2, ..., n
What are the sums of the powers in each term equal to?
The sum of the powers of a and b in each term is always n (the power of the bracket before expansion)
What is the type of pattern of the coefficients for each term?
The pattern of the coefficients (the numbers in front of the letters) in each line is symmetrical
What are the steps when we want to expand and simplify (x+2)^5?
1: write the x to the power values decending: x^5 + x^4 + x^3 + x^2 + x^1 2: write in the y to the power values (2) ascending, starting from the second term: x^5 + x^4(2)^1 + x^3(2)^2 + x^2(2)^3 + x^1(2)^4 + (2)^5 3: write in the coefficients from memory from pascals triangle or nCr on the calculator: 1x^5 + 5x^4(2)^1 + 10x^3(2)^2 + 10x^2(2)^3 + 5x^1(2)^4 + 1(2)^5 4: Multiply them out and collect like terms: = x^5 + 10x^4 + 40x^3 + 80x^2 + 80x + 32
What is meant by 4!?
4 factorial = 4×3×2×1=24
What are the numbers infront of the terms called?
Binomial coefficients
How can we find bionmial coefficients on the calculator?
Binomial coefficients can be determined using the nCr button on your calculator, where n is the power the bracket is being raised to and r is the position of the term in the expansion starting with r=0.
How would we expand (2x+1)(x+3)^5?
First use the binomial theorem to expand the last bracket (x+3)^5, then multiply all the values first by 2x and then 1, then bring the like terms together.
In the binomial sequence for (a+b)^n, what happens to the powers of a in the sequence?
In each expression, the power of a starts at n and decreases by 1 for each term, so the powers are n,n−1,n−2, ...,0
What can we use to find these bionomial coefficeints for low numbers of n?
Pascals triangle
What does binomial theorem look at?
The binomial theorem looks at the result of raising a single bracket containing two terms to a positive integer power. In other words, a binomial expansion involves multiplying the same bracket by itself a given number of times.
What do you do when the terms in the original have coefficients larger than 1, for example: (2x+3)^4?
The coefficient is treated as part of the term being raised to the power, for example: 1(2x)^4 + 4(2x)^3(3) + 6(2x)^2(3^2) + 4(2x)(3^3) + 3^4
How many terms are there in each expression relative to n?
The number of terms in each expansion is always one more than the power, n.
What must you also do when there is a minus sign as a coeficcient?
Use brackets to make clear that the minus sign is also being raised to the power in each term.
How can we use bionmial theorem to approximate the value of (0.97)^7?
We could write this as (1−3x)^7 where x is 0.01. Substituting x=0.01 into the expanded form of (1−3x)^7 would then give us the value of (0.97)^7. However, since larger powers of 0.01 will be very close to zero, we can ignore them. Work out the binomial sequence for (1−3x)^7, then substitute 0.01 in for x to find an apporixamation of the figure.
What is the second and second to last coefficeint in the bionmial series?
n
What is nCr read as?
n choose r
What is the formula for the bionomial coefficient of the nCr'th term?
n!/r!(n-r)! where n is the power and r is the chosen term in the bionmial sequence
What is ℕ?
ℕ is the set of natural numbers (or positive integers): 0, 1, 2, 3, 4, 5...)
