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The Binomial Expansion

The Binomial Expansion. Introduction. You first met the Binomial Expansion in C2 In this chapter you will have a brief reminder of expanding for positive integer powers We will also look at how to multiply out a bracket with a fractional or negative power

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The Binomial Expansion

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  1. The Binomial Expansion

  2. Introduction • You first met the Binomial Expansion in C2 • In this chapter you will have a brief reminder of expanding for positive integer powers • We will also look at how to multiply out a bracket with a fractional or negative power • We will also use partial fractions to allow the expansion of more complicated expressions

  3. Teachings for Exercise 3A

  4. The Binomial Expansion You need to be able to expand expressions of the form (1 + x)n where n is any real number Find: Always start by writing out the general form Sub in: n = 4 x = x Work out each term separately and simplify Every term after this one will contain a (0) so can be ignored  The expansion is finite and exact 3A

  5. The Binomial Expansion You need to be able to expand expressions of the form (1 + x)n where n is any real number Find: Always start by writing out the general form Sub in: n = 3 x = -2x Work out each term separately and simplify It is VERY important to put brackets around the x parts Every term after this one will contain a (0) so can be ignored  The expansion is finite and exact 3A

  6. The Binomial Expansion You need to be able to expand expressions of the form (1 + x)n where n is any real number Find: Rewrite this as a power of x first Write out the general form (it is very unlikely you will have to go beyond the first 4 terms) Sub in: n = -1 x = x Work out each term separately and simplify • With a negative power you will not get a (0) term • The expansion is infinite • It can be used as an approximation for the original term 3A

  7. The Binomial Expansion You need to be able to expand expressions of the form (1 + x)n where n is any real number Find: Rewrite this as a power of x first Write out the general form (it is very unlikely you will have to go beyond the first 4 terms) Sub in: n = 1/2 x = -3x Work out each term separately and simplify  You should use your calculator carefully • With a fractional power you will not get a (0) term • The expansion is infinite • It can be used as an approximation for the original term 3A

  8. The Binomial Expansion You need to be able to expand expressions of the form (1 + x)n where n is any real number and state the values of x for which it is valid… Find the Binomial expansion of: Write out the general form Sub in: n = 1/3 x = -x Work out each term separately and simplify Imagine we substitute x = 2 into the expansion The values fluctuate (easier to see as decimals)  The result is that the sequence will not converge and hence for x = 2, the expansion is not valid 3A

  9. The Binomial Expansion You need to be able to expand expressions of the form (1 + x)n where n is any real number and state the values of x for which it is valid… Find the Binomial expansion of: Write out the general form Sub in: n = 1/3 x = -x Work out each term separately and simplify Imagine we substitute x = 0.5 into the expansion The values continuously get smaller  This means the sequence will converge (like an infinite series) and hence for x = 0.5, the sequence IS valid… 27 3A

  10. The Binomial Expansion You need to be able to expand expressions of the form (1 + x)n where n is any real number and state the values of x for which it is valid… Find the Binomial expansion of: Write out the general form Sub in: n = 1/3 x = -x Work out each term separately and simplify How do we work out for what set of values x is valid? The reason an expansion diverges or converges is down to the x term… If the term is bigger than 1 or less than -1, squaring/cubing etc will accelerate the size of the term, diverging the sequence If the term is between 1 and -1, squaring and cubing cause the terms to become increasingly small, to the sum of the sequence will converge, and be valid Write using Modulus The expansion is valid when the modulus value of x is less than 1 3A

  11. The Binomial Expansion You need to be able to expand expressions of the form (1 + x)n where n is any real number and state the values of x for which it is valid… Find the Binomial expansion of: Write out the general form: Sub in: n = -2 x = 4x Work out each term separately and simplify The ‘x’ term is 4x… Divide by 4 3A

  12. The Binomial Expansion You need to be able to expand expressions of the form (1 + x)n where n is any real number and by using x = 0.01, find an estimate for √2 Find the Binomial expansion of: Write out the general form: Sub in: n = 1/2 x = -2x Work out each term separately and simplify 3A

  13. The Binomial Expansion You need to be able to expand expressions of the form (1 + x)n where n is any real number and by using x = 0.01, find an estimate for √2 Find the Binomial expansion of: x = 0.01 Rewrite left using a fraction Square root top and bottom separately Multiply by 10 Divide by 7 3A

  14. Teachings for Exercise 3B

  15. The Binomial Expansion You can use the expansion for (1 + x)n to expand (a + bx)n by taking out a as a factor Find the first 4 terms in the Binomial expansion of: Take a factor 4 out of the brackets Both parts in the square brackets are to the power 1/2 You can work out the part outside the bracket Write out the general form: Sub in: n = 1/2 x = x/4 Work out each term carefully and simplify it Remember we had a 2 outside the bracket  Multiply each term by 2 Multiply by 4 3B

  16. The Binomial Expansion You can use the expansion for (1 + x)n to expand (a + bx)n by taking out a as a factor Find the first 4 terms in the Binomial expansion of: Take a factor 2 out of the brackets Both parts in the square brackets are to the power -2 You can work out the part outside the bracket Write out the general form: Sub in: n = -2 x = 3x/2 Work out each term carefully and simplify it Remember we had a 1/4 outside the bracket  Divide each term by 4 Multiply by 2, divide by 3 3B

  17. Teachings for Exercise 3C

  18. The Binomial Expansion You can use Partial fractions to simplify the expansions of more difficult expressions Find the expansion of: up to and including the term in x3 Express as Partial Fractions Cross-multiply and combine The numerators must be equal If x = 2 If x = -1 Express the original fraction as Partial Fractions, using A and B 3C

  19. The Binomial Expansion You can use Partial fractions to simplify the expansions of more difficult expressions Find the expansion of: up to and including the term in x3 Both fractions can be rewritten Expand each term separately Write out the general form: Sub in: x = x n = -1 Work out each term carefully Remember that this expansion is to be multiplied by 3 3C

  20. The Binomial Expansion You can use Partial fractions to simplify the expansions of more difficult expressions Find the expansion of: up to and including the term in x3 Both fractions can be rewritten Expand each term separately Take a factor 2 out of the brackets (and keep the current 2 separate…) Both parts in the square brackets are raised to -1 Work out 2-1 This is actually now cancelled by the 2 outside the square bracket! 3C

  21. The Binomial Expansion You can use Partial fractions to simplify the expansions of more difficult expressions Find the expansion of: up to and including the term in x3 Both fractions can be rewritten Expand each term separately Write out the general form: Sub in: x = -x/2 n = -1 Work out each term carefully 3C

  22. The Binomial Expansion You can use Partial fractions to simplify the expansions of more difficult expressions Find the expansion of: up to and including the term in x3 Both fractions can be rewritten Replace each bracket with its expansion Subtract the second from the first (be wary of double negatives in some questions) 3C

  23. Summary • We have been reminded of the Binomial Expansion • We have seen that when the power is a positive integer, the expansion is finite and exact • With negative or fractional powers, the expansion is infinite • We have seen how to decide what set of x-values the expansion is valid for • We have also used partial fractions to break up more complex expansions

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