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Home/Tag: Binomials
30 03, 2018

## Completing the Square

Completing the Square Half and square worked examples Complete the square for x2 + 6x. = x2 + 6x  + 9 - 9 consider the coefficient of the x term: 6 find half of 6 = 3 and then square it 32 = 9 and add 9 to the [...]

9 07, 2017

## Finding Coefficients

Finding Coefficients ${}^n{C_r}{(a)^{n - r}}{(b)^r} = A{x^c}$   worked examples Find the coefficient of x4 in the expansion of  ${\left( {x - \frac{2}{x}} \right)^{12}}$. 12Ck x12 – k (2/x)k = Ax4 using the formula ${}^n{C_r}{(a)^{n - r}}{(b)^r} = A{x^c}$ where n = 12, a =  x, b = [...]

9 07, 2017

## Independent Term

Independent Term ${}^n{C_r}{(a)^{n - r}}{(b)^r} = A{x^0}$   worked examples Find the term that is independent of x in the expansion of ${\left( {\frac{1}{{{x^3}}} + 2{x^5}} \right)^{16}}$.   ${}^{16}{C_k}{\left( {\frac{1}{{{x^3}}}} \right)^{16 - k}}{(2{x^5})^k} = A{x^0}$   using the formula ${}^n{C_r}{(a)^{n - r}}{(b)^r} = A{x^0}$ where n = 16, a [...]

27 04, 2017

## Pascal’s Triangle

Pascal's Triangle To construct a Pascal's Triangle, start with 1 at the top, a single number All rows must start and end with 1, so the second row will have two numbers : 1 and 1 The third row - first number is 1, to get the next number, you [...]

31 05, 2016

Quadratic Factors - Harder Case worked examples Factorise 4x2 + 4x - 15.  Method 1 4x2 + 4x - 15 = 4x2 + 10x - 6x - 15 Step 1: multiply the coefficient of x2 by the constant ie 4 × -15 = -60 Step 2: Consider what will [...]

19 08, 2015

## Challenging Proofs

Challenging Binomial Proofs Consider the expansion 2nCo + 2nC1x + 2nC2x2 + … +2nC2nx2n = (1 + x)2n. Show that 2nCo + 2nC1 + 2nC1 + … + 2nCn = ${2^{2n - 1}} + \frac{{(2n)!}}{{2{{(n!)}^2}}}$ 2nCo + 2nC1x + 2nC2x2 + … + 2nCn – 1 xn – 1 [...]

18 08, 2015

## General Binomial Expansion

General Binomial Expansion an - bn = (a - b)(an-1 + an-2b + an-3b2 +... abn-2 + bn-1)   worked examples further examples = 10x + 15 - 7x + 14 = 3x + 29 harder examples [...]

18 08, 2015

## Coefficient Relationships

Coefficient Relationships worked examples The coefficients of x10 and x5 are equal in the expansion of (1 + x)n. Find the value of n.   consider the terms nC8x8 and nC7x7 and look only at their coefficients $\frac{n!}{8!(n-8)!}=\frac{n!}{7!(n-7)!}$ divide both sides by n! $\frac{1}{8!(n-8)!}=\frac{1}{7!(n-7)!}$ reciprocal both sides  8!(n - 8)! = 7!(n [...]

18 08, 2015

## Greatest Coefficient

Greatest Coefficient $\frac{{{T_{k + 1}}}}{{{T_k}}} > 1$ $\frac{n-k+1}{k}.\frac{b}{a}>1$ worked examples Find the greatest coefficient of (x2 + 2)9.   $\frac{{{}^9{C_k}{{({x^2})}^{9 - k}}{{(2)}^k}}}{{{}^9{C_{k - 1}}{{({x^2})}^{9 - (k - 1)}}{{(2)}^{k - 1}}}} > 1$   using the formula $\frac{{{T_{k + 1}}}}{{{T_k}}} > 1$ where, nCk = $\frac{{n!}}{{k!(n - k)!}}$ Tk + [...]