expand (1-x^2)^6
-The signs of an (a-b)^n will alternate positive, then negative, etc.
-recall pascal's triangle; this will help you with the coefficients
this website will help guide you through the problem:
algebralab[dot]org[forward slash]lessons[forward slash]lesson[dot]aspx?file=Algebra_BinomialExpansion.xml
(1 -X^2)^6 =
(1 - X^2)^2(1 - X^2)^2(1 - X^2)^2(1 -X^2),
(X^4 - 2X^2 + 1)(X^4 - 2X^2 +1)(X^4 - 2X + 1),
(X^8 - 2X^6 + X^4 - 2X^6 + 4X^4 -2X^2 + X^4 - 2X^2 +1)(X^4 - 2X^2 + 1)
Combine like-terms:
(X^4 - 2X^2 + 1)(X^8 - 4X^6 + 6X^4 -4X^2 + 1),
Multiply to remove parenthesis:
X^12 - 4X^10 + 6X^8 - 4X^6 + X^4
- 2X^10 + 8X^8 - 12X^6 + 8X^4 - 2X^2
+ X^8 - 4X^6 + 6X^4 - 4X^2 + 1
Combine like-terms:
X^12 - 6X^10 +15X^8 - 20X^6 + 15X^4
- 6X^2 + 1.
Since our given binomial was raised to the 6th power, our final results should be 7 terms. Always 1 greater
than the exponential.
To expand the expression (1 - x^2)^6, we can use the binomial expansion theorem. According to the theorem, when we expand an expression of the form (a + b)^n, the result will be a sum of terms, each of which is obtained by multiplying a term from the first binomial (a) by a term from the second binomial (b), where the sum of their exponents is equal to n.
In our case, the first binomial is 1, and the second binomial is (-x^2). The exponent is 6.
To expand (1 - x^2)^6, we can use the binomial formula:
(n choose k) * a^(n-k) * b^k
Where (n choose k) is the binomial coefficient, a is the coefficient of the first term, b is the coefficient of the second term, n is the exponent, and k is the power to which the second term is raised.
Let's apply this formula to (1 - x^2)^6:
(6 choose 0) * 1^(6-0) * (-x^2)^0 +
(6 choose 1) * 1^(6-1) * (-x^2)^1 +
(6 choose 2) * 1^(6-2) * (-x^2)^2 +
(6 choose 3) * 1^(6-3) * (-x^2)^3 +
(6 choose 4) * 1^(6-4) * (-x^2)^4 +
(6 choose 5) * 1^(6-5) * (-x^2)^5 +
(6 choose 6) * 1^(6-6) * (-x^2)^6
Simplifying the terms:
1 +
6 * (-x^2) +
15 * (x^4) +
20 * (-x^6) +
15 * (x^8) +
6 * (-x^10) +
1 * (x^12)
The expansion of (1 - x^2)^6 is:
1 - 6x^2 + 15x^4 - 20x^6 + 15x^8 - 6x^10 + x^12