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What do you notice?
-The numbers in each column down from the first is the sum of the two numbers above it.
This is important, because to expand a binomial, we need this property.
(x^2 - 2)^4 => 5(x2−2)4⇒5 terms, since to find the number of terms you always add 1 to the exponent.
For 5 terms, you must pick the row with 5 terms in the Pascal's Triangle. The row with 5 terms is 1, 4, 6, 4 and 1.
We must multiply these numbers with exponents starting at 4 and ending at 0 for x^2x2 and start at 0 and ending at 4 for -2−2. We do that since the total exponent of the expression is 4, and if you include the 0th term, you have your 5 terms.
(x^2 - 2)^4 = 1(x^2)^4(-2)^0 + 4(x^2)^3(-2)^1 + 6(x^2)^2(-2)^2 + 4(x^2)^1(-2)^3 + 1(x^2)^0(-2)^4(x2−2)4=1(x2)4(−2)0+4(x2)3(−2)1+6(x2)2(−2)2+4(x2)1(−2)3+1(x2)0(−2)4
=x^8(1) + 4(x^6)(-2) + 6(x^4)(4) + 4(x^2)(-8) + 16x8(1)+4(x6)(−2)+6(x4)(4)+4(x2)(−8)+16
=x^8 - 8x^6 + 24x^4 - 32x^2 + 16x8−8x6+24x4−32x2+16
So, (x^2 - 2)^4 = x^8 - 8x^6 + 24x^4 - 32x^2 + 16(x2−2)4=x8−8x6+24x4−32x2+16
Practice exercises:
- Expand the following using Pascal's Triangle.
a) (2x - 5y)^5(2x−5y)5
b) (y^2 + 2x^2)^7(y2+2x2)7
Challenge problem:
Expand (2x - 3y)^-4(2x−3y)−4 using Pascal's Triangle. Hint: remember the exponent rule about how to turn a negative exponent into a positive exponent!
Good luck!
