%5E4.jpeg "(2x+1)^4")
Expanding (2x + 1)^4
Expanding expressions like (2x + 1)^4 can be done in a couple of ways:
1. Binomial Theorem
The binomial theorem provides a general formula for expanding expressions of the form (a + b)^n.
The Formula:
(a + b)^n = n!/(0!n!)a^nb^0 + n!/(1!(n-1)!)*a^(n-1)*b^1 + ... + n!/(n!0!)a^0b^n
Applying it to (2x + 1)^4:
- a = 2x
- b = 1
- n = 4
Substituting these values into the binomial theorem, we get:
(2x + 1)^4 = 4!/(0!4!)(2x)^41^0 + 4!/(1!3!)(2x)^31^1 + 4!/(2!2!)(2x)^21^2 + 4!/(3!1!)(2x)^11^3 + 4!/(4!0!)(2x)^01^4
Simplifying:
(2x + 1)^4 = 16x^4 + 32x^3 + 24x^2 + 8x + 1
2. Repeated Multiplication
We can also expand (2x + 1)^4 by repeatedly multiplying the expression by itself:
(2x + 1)^4 = (2x + 1)(2x + 1)(2x + 1)(2x + 1)
-
Expand the first two terms: (2x + 1)(2x + 1) = 4x^2 + 4x + 1
-
Multiply the result by (2x + 1): (4x^2 + 4x + 1)(2x + 1) = 8x^3 + 12x^2 + 6x + 1
-
Multiply the result by (2x + 1): (8x^3 + 12x^2 + 6x + 1)(2x + 1) = 16x^4 + 32x^3 + 24x^2 + 8x + 1
Therefore, we arrive at the same result: (2x + 1)^4 = 16x^4 + 32x^3 + 24x^2 + 8x + 1
Conclusion
Both methods, the binomial theorem and repeated multiplication, lead to the same expanded form of (2x + 1)^4. The binomial theorem is more efficient for larger exponents, while repeated multiplication can be more intuitive for understanding the process.