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/Tag: Binomials
30 03, 2018

## Completing the Square

By |2018-03-30T11:01:35+10:00March 30th, 2018||0 Comments

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

By |2017-07-09T17:03:48+10:00July 9th, 2017||0 Comments

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

By |2017-07-09T09:47:57+10:00July 9th, 2017||0 Comments

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

By |2017-04-27T13:00:04+10:00April 27th, 2017||0 Comments

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

## Harder Quadratic Factors

By |2016-11-01T10:59:33+10:00May 31st, 2016||0 Comments

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

By |2016-11-08T16:59:29+10:00August 19th, 2015||0 Comments

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

By |2016-11-08T16:58:17+10:00August 18th, 2015||0 Comments

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

By |2018-02-11T12:19:49+10:00August 18th, 2015||0 Comments

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

By |2017-07-08T23:20:02+10:00August 18th, 2015||0 Comments

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 + [...]

18 08, 2015

## Binomials

By |2017-07-09T17:05:43+10:00August 18th, 2015|Tags: , |0 Comments

Binomials Go to the Binomials Forum to see more questions or ask your own Topics Covered in Binomials Pascal's Triangle Greatest Coefficient Finding Coefficients Independent Term Coefficient Relationships Binomial Expansion Challenging Proofs   Formulas Used in [...]