%5E10%2B(x%2B2)%5E10.jpeg "(x+1)^10+(x+2)^10")
Expanding the Expression: (x+1)^10 + (x+2)^10
In this article, we will explore the expansion of the expression (x+1)^10 + (x+2)^10. This expression involves the sum of two binomials, each raised to the power of 10. We will use the binomial theorem to expand each term and then combine like terms to simplify the expression.
Binomial Theorem
The binomial theorem states that for any positive integer n:
$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$
where n is a positive integer, a and b are real numbers, and \binom{n}{k} is the binomial coefficient.
Expanding (x+1)^10
Using the binomial theorem, we can expand (x+1)^10 as:
$(x+1)^{10} = \sum_{k=0}^{10} \binom{10}{k} x^{10-k} 1^k$
Simplifying the expression, we get:
$(x+1)^{10} = x^{10} + 10x^9 + 45x^8 + 120x^7 + 210x^6 + 252x^5 + 210x^4 + 120x^3 + 45x^2 + 10x + 1$
Expanding (x+2)^10
Using the binomial theorem, we can expand (x+2)^10 as:
$(x+2)^{10} = \sum_{k=0}^{10} \binom{10}{k} x^{10-k} 2^k$
Simplifying the expression, we get:
$(x+2)^{10} = x^{10} + 20x^9 + 180x^8 + 640x^7 + 1680x^6 + 2800x^5 + 3360x^4 + 3136x^3 + 1792x^2 + 512x + 1024$
Combining the Two Expansions
Now, we can combine the two expansions to get:
$(x+1)^{10} + (x+2)^{10} = 2x^{10} + 30x^9 + 225x^8 + 760x^7 + 1890x^6 + 3052x^5 + 4176x^4 + 4992x^3 + 4304x^2 + 1536x + 1025$
Therefore, the expansion of (x+1)^10 + (x+2)^10 is a polynomial of degree 10 with 11 terms.
In conclusion, we have successfully expanded the expression (x+1)^10 + (x+2)^10 using the binomial theorem. The resulting polynomial has 11 terms and can be used for further mathematical manipulations.
