`2(x - 3)^4 + 5(x - 3)^2` Use the Binomial Theorem to expand and simplify the expression.

Expert Answers
sciencesolve eNotes educator| Certified Educator

 

 

You need first to factorize `(x - 3)^2` , such that:

`2(x-3)^4 + 5(x-3)^2 = (x-3)^2(2(x-3)^2 + 5)`

You need to use the binomial formula, such that:

`(x+y)^n = sum_(k=0)^n ((n),(k)) x^(n-k) y^k`

You need to replace x for x, 3for y and 2 for n, such that:

`(x-3)^2 = 2C0 (x)^2+2C1 (x)^1*(-3)^1+2C2(-3)^2`

By definition, nC0 = nCn = 1, hence `2C0 = 2C2 = 1.`

By definition `nC1 = nC(n-1) = n` , hence `2C1= 2.`

`(x-3)^2 = x^2 - 6x + 9`

Replacing the binomial expansion` x^2 - 6x + 9` for `(x-3)^2`  yields:

`(x-3)^2(2(x-3)^2 + 5) = (x^2 - 6x + 9)(2(x^2 - 6x + 9) + 5)`

`(x-3)^2(2(x-3)^2 + 5) = (x^2 - 6x + 9)(2x^2 - 12x +18 + 5)`

`(x-3)^2(2(x-3)^2 + 5) = (x^2 - 6x + 9)(2x^2 - 12x +23)`

`(x-3)^2(2(x-3)^2 + 5) = (2x^4 - 12x^3 + 23x^2 - 12x^3 + 72x^2 - 138x + 18x^2 - 108x + 207)`

Grouping the like terms yields:

`2(x-3)^4 + 5(x-3)^2 = 2x^4 - 24x^3 + 113x^2 - 246x + 207`

Hence, expanding and simplifying the expression yields `2(x-3)^4 + 5(x-3)^2 = 2x^4 - 24x^3 + 113x^2 - 246x + 207.`

gsarora17 eNotes educator| Certified Educator

Let's use the binomial formula, `(x+y)^n=sum_(k=0)^n ((n),(k))(x)^(n-k)*y^k` to expand `2(x-3)^4+5(x-3)^2`

`2(x-3)^4=2(((4),(0)).x^(4-0)*(-3)^0+((4),(1))*x^(4-1)*(-3)^1+((4),(2))*x^(4-2)*(-3)^2+((4),(3))*x^(4-3)*(-3)^3+((4),(4))*x^(4-4)*(-3)^4)`

`=2(x^4+(4!)/(1!(4-1)!)*x^3*(-3)+(4!)/(2!(4-2)!)*x^2*(-3)^2+(4!)/(3!(4-3)!)*x^1*(-3)^3+(-3)^4)`

`=2(x^4+(4*3!)/(3!)*x^3*(-3)+(4*3*2!)/(2!2!)*x^2*(-3)^2+(4*3!)/(3!1!)*x*(-3)^3+(-3)^4)`

`=2(x^4-12x^3+54x^2-108x+81)`

`5(x-3)^2=5(((2),(0))*x^(2-0)*(-3)^0+((2),(1))*x^(2-1)*(-3)^1+((2),(2))*x^(2-2)*(-3)^2)`

`=5(x^2+(2!)/(1!(2-1)!)*x^1*(-3)^1+(-3)^2)`

`=5(x^2-6x+9)`

`:.2(x-3)^4+5(x-3)^2=2(x^4-12x^3+54x^2-108x+81)+5(x^2-6x+9)`

`=2x^4-24x^3+108x^2-216x+162+5x^2-30x+45`

Now combine the like terms,

`=2x^4-24x^3+108x^2+5x^2-216x-30x+162+45`

`=2x^4-24x^3+113x^2-246x+207`