Finding Binomial Expansion of $(1-3x)^7$
This Binomial Expansion Calculator is helpful for Finding Binomial Expansion of (2+h) 3 . We can easily calculate the value of an expression having lower power but the binomial theorem can fine binomial expansion for any expression. In this article, we will find the Binomial Expansion of (2+h) 3 in some easy and quick steps.
Ex: (x+1)^2 (or) (x+7)^7 (or) (x+3)^4
Elaborate Steps to Expand $(1-3x)^7$ Using Binomial Theorem
Given expression is $(1 - 3*x)^7$
The binomial Theorem says that to expand any non-negative power of binomial (x+y), then use the below formula,
=> (x+y)n = nC0xny0 +nC1xn-1y1 + nC2xn-2y2 + ... + nCn-1x1yn-1 +nCnx0yn
$(1 - 3.x)^7$ = $\\sum_{k=0}^7 {^7C_k}((1)^{7-k}(-3.x)^{k})$
By expansion,
$(1 - 3*x)^7$=
$\frac{7!}{(7-0)!0!}(1)^{7-0}\times{}(-3.x)^0+\frac{7!}{(7-1)!1!}(1)^{7-1}\times{}(-3.x)^1+\frac{7!}{(7-2)!2!}(1)^{7-2}\times{}(-3.x)^2+\frac{7!}{(7-3)!3!}(1)^{7-3}\times{}(-3.x)^3+\frac{7!}{(7-4)!4!}(1)^{7-4}\times{}(-3.x)^4+\frac{7!}{(7-5)!5!}(1)^{7-5}\times{}(-3.x)^5+\frac{7!}{(7-6)!6!}(1)^{7-6}\times{}(-3.x)^6+\frac{7!}{(7-7)!7!}(1)^{7-7}\times{}(-3.x)^7$
$= \frac{5040}{(5040)1}(1)^{7-0}\times{}(-3.x)^0+\frac{5040}{(720)1}(1)^{7-1}\times{}(-3.x)^1+\frac{5040}{(120)2}(1)^{7-2}\times{}(-3.x)^2+\frac{5040}{(24)6}(1)^{7-3}\times{}(-3.x)^3+\frac{5040}{(6)24}(1)^{7-4}\times{}(-3.x)^4+\frac{5040}{(2)120}(1)^{7-5}\times{}(-3.x)^5+\frac{5040}{(1)720}(1)^{7-6}\times{}(-3.x)^6+\frac{5040}{(1)5040}(1)^{7-7}\times{}(-3.x)^7$
$= 1(1)^{7-0}\times{}(-3.x)^0+7(1)^{7-1}\times{}(-3.x)^1+21(1)^{7-2}\times{}(-3.x)^2+35(1)^{7-3}\times{}(-3.x)^3+35(1)^{7-4}\times{}(-3.x)^4+21(1)^{7-5}\times{}(-3.x)^5+7(1)^{7-6}\times{}(-3.x)^6+1(1)^{7-7}\times{}(-3.x)^7$
$= (1)^{7-0}\times{}(-3.x)^0+(7)(1)^{7-1}\times{}(-3.x)^1+(21)(1)^{7-2}\times{}(-3.x)^2+(35)(1)^{7-3}\times{}(-3.x)^3+(35)(1)^{7-4}\times{}(-3.x)^4+(21)(1)^{7-5}\times{}(-3.x)^5+(7)(1)^{7-6}\times{}(-3.x)^6+(1)^{7-7}\times{}(-3.x)^7$
$= (1)^{7}\times{}(-3.x)^0+(7)(1)^{6}\times{}(-3.x)^1+(21)(1)^{5}\times{}(-3.x)^2+(35)(1)^{4}\times{}(-3.x)^3+(35)(1)^{3}\times{}(-3.x)^4+(21)(1)^{2}\times{}(-3.x)^5+(7)(1)^{1}\times{}(-3.x)^6+(1)^{0}\times{}(-3.x)^7$
$= (1)^{7}\times{}1+(7)(1)^{6}\times{}(-3.x)^1+(21)(1)^{5}\times{}(-3.x)^2+(35)(1)^{4}\times{}(-3.x)^3+(35)(1)^{3}\times{}(-3.x)^4+(21)(1)^{2}\times{}(-3.x)^5+(7)(1)^{1}\times{}(-3.x)^6+1\times{}(-3.x)^7$
$= 1\times{}(1)+(7)1\times{}(-3.x)+(21)1\times{}(9.x^2)+(35)1\times{}(-27.x^3)+(35)1\times{}(81.x^4)+(21)1\times{}(-243.x^5)+(7)1\times{}(729.x^6)+1\times{}(-2187.x^7)$
$= -2187x^7 + 5103x^6 - 5103x^5 + 2835x^4 - 945x^3 + 189x^2 - 21x + 1$
Therefore, the binomial expansion of $(1 - 3.x)^7$ is $-2187x^7 + 5103x^6 - 5103x^5 + 2835x^4 - 945x^3 + 189x^2 - 21x + 1$
FAQs on Binomial Expansion of $(1-3x)^7$
1. How many terms are in the binomial expression of $(1-3x)^7$?
The number of terms in the binomial expansion of $(1-3x)^7$ is 8
2. How to find the binomial expansion of $(1-3x)^7$ ?
We use the Binomial theorem to find the expansion of $(1-3x)^7$ . The formula is (x + y)n = Σr=0n (nCr xn – ryr).
3. What is the binomial expansion of $(1-3x)^7$ ?
The binomial expansion of $(1-3x)^7$ is $-2187x^7 + 5103x^6 - 5103x^5 + 2835x^4 - 945x^3 + 189x^2 - 21x + 1$ .