I need to use Pascal's Triangle to expand (x-1)^5
I know I use the fifth row but I'm not sure how to do it- I have a bunch of these and I'm really confused- Please help if you can-
you know the expansion must look like
?x^5(-1)^0 + ?x^4(-1)^1 + ?x^3(-1)^2 + ?x^2(-1)^3 + ?x^1(-1)^4 + ?x^0(-1)^5
= ?x^5 - ?x^4 + ?x^3 - ?x^2 + ?x - ?
now the 5th row of the triangle has values:
1 5 10 10 5 1
put these in where the ?'s are and you are done
= x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1
suppose you had
(x+2)^4
= 1x^4 + 4x^3(2) + 6x^2(2^2) + 4x(2^3) + 12^4
= x^4 + 8x^3 + 24x^2 + 32x + 16
That last term in the 2nd last row of course is 1(2^4) or 16
Certainly! To expand (x - 1)^5 using Pascal’s Triangle, you can follow these steps:
Step 1: Identify the fifth row in Pascal’s Triangle. The rows are usually numbered starting from 0, so the fifth row will be row number 4.
Step 2: Write down the coefficients of the terms in the fifth row. In Pascal’s Triangle, each number is obtained by adding the two numbers directly above it. So, the fifth row of Pascal’s Triangle goes as follows: 1, 4, 6, 4, 1.
Step 3: Rewrite the expression (x - 1)^5 using the coefficients from the fifth row. This can be done by raising each part of the binomial, (x - 1), to the corresponding power and then multiplying by the respective coefficient.
In this case, the expression (x - 1)^5 will be expanded as follows:
(1 * x^5) + (4 * x^4 * -1^1) + (6 * x^3 * -1^2) + (4 * x^2 * -1^3) + (1 * x^1 * -1^4)
Remember that any number raised to an even power gives a positive result, and any number raised to an odd power gives a negative result.
Simplifying the above expression will give you the expanded form of (x - 1)^5.
Let me know if you would like a step-by-step breakdown of the actual calculations, or if you have any further questions!