Expand (3-x)(2+x)3 by using a binomial expansion
To expand the expression (3-x)(2+x)3 using a binomial expansion, we can use the formula for expanding a binomial raised to a power.
The formula for expanding (a+b)^3 is:
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
In this case, a = (3-x) and b = (2+x).
Using the formula, the expanded expression is:
(3-x)^3 + 3(3-x)^2(2+x) + 3(3-x)(2+x)^2 + (2+x)^3
Now let's expand each term:
(3-x)^3 = (3-x)(3-x)(3-x) = (9 - 6x + x^2)(3-x) = 27 - 45x + 22x^2 - 3x^3
3(3-x)^2(2+x) = 3(9 - 6x + x^2)(2+x) = 3(18 - 18x + 3x^2 + 2x + x^2) = 54 - 72x + 18x^2 + 6x^2 + 3x^3
3(3-x)(2+x)^2 = 3(3-x)(4 + 4x + x^2) = 3(12 - 12x + 3x^2 + 4x + 4x^2 + x^3) = 36 - 36x + 9x^2 + 12x + 12x^2 + 3x^3
(2+x)^3 = (2+x)(2+x)(2+x) = (4 + 4x + x^2)(2+x) = 8 + 12x + 6x^2 + 4x + 2x^2 + x^3
Now, combining like terms, we get:
27 - 45x + 22x^2 - 3x^3 + 54 - 72x + 18x^2 + 6x^2 + 3x^3 + 36 - 36x + 9x^2 + 12x + 12x^2 + 3x^3 + 8 + 12x + 6x^2 + 4x + 2x^2 + x^3
Simplifying further, we get:
27 - 45x + 22x^2 - 3x^3 + 54 - 72x + 18x^2 + 6x^2 + 3x^3 + 36 - 36x + 9x^2 + 12x + 12x^2 + 3x^3 + 8 + 12x + 6x^2 + 4x + 2x^2 + x^3
Now, combining like terms, we get:
27 - 45x + 22x^2 - 3x^3 + 54 - 72x + 24x^2 + 9x^3 + 36 - 36x + 21x^2 + 12x + 12x^2 + 3x^3 + 8 + 12x + 9x^2 + 4x + 2x^2 + x^3
Simplifying further, we get:
27 + 54 + 36 + 8 + (-45x - 72x - 36x - 36x - 12x) + (22x^2 + 24x^2 + 21x^2 + 9x^2) + (-3x^3 + 9x^3 + 3x^3 + x^3)
Combining like terms, we get the final expanded expression:
125 - 228x + 65x^2 + 14x^3.