THE BINOMIAL THEOREM

PROOF FOR POSITIVE INTEGRAL EXPONENTS

489. The Binomial Theorem has already been found inductively, and the laws relating to the exponents and coefficients of the letters in the formula by which it is expressed have been stated (Arts. 119, 264). A proof of the theorem for positive integral exponents will now be given.

490. THEOREM. If the binomial a + b is raised to the nth power, n being a positive integer, the result is expressed by the formula

I. The theorem is evidently true for n = 1, since

(2)

It is also true for the 2d power, since by multiplication

(a+b)2= a2 + 2 ab + b2

a2b0 +2 a1b1 + aob2.

By multiplication we also know that

(a + b)3 = a3 +3 a2b + 3 ab2 + b3

= a3bo+3a2b1 +3 a1b2 + aob3.

(3)

We

II. Assume, for the present, that the theorem is true when n has the particular value r; that is, for the rth power. shall then have

Multiplying both members of (4) by a + b, we have

(4)

2

Upon comparison this result is seen to be of the same form

as the rth power, for replacing

by r+1 in the second mem

which is identical with the second member of (5), and hence is

identical with (a + b)r+1.

Hence it is seen that if the theorem were true for the rth power, it would be true for the (r+1)th power:

III. But the theorem has been shown to be true for the 1st, 2d, and 3d powers, by step I. Therefore, by II, it must be true for the (3+1)th, or 4th power, and therefore for the (4+1)th, or 5th power; and so on for all positive integral powers.

It can be proved that the theorem is true when n is negative or fractional, but the proof is too difficult for this stage.

491. The method of proof employed in the last article is called Mathematical Induction. It is to be noticed that there are two distinct parts of the proof as applied above. In the one step, we are to show that, if the theorem is true for any particular value of n, as n = r, then it will also be true for the next value n =r+1. In the other step, it is necessary to show that the theorem is true for some initial value of n, as for n = n = 2, or n = 3, so that there may be a starting-point free from "ifs" from which to pass by the induction process to the next higher power, and from this to the next higher power, and

so on.

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492. The expansion of (a - b)" can be written from that of (a+b)" by noting that

Observe that when b is negative the terms containing even powers of b are positive, and the terms containing odd powers are negative.

493. Useful application of the Binomial Theorem has already been made in raising binomials to various powers, but a few exercises are here given for additional practice. By assuming that the theorem is true, whether n is positive, negative, integral, or fractional, the exercises are extended to include the expansion of negative and fractional exponents.

1. Expand (1 + x)✯ to 3 terms.

Comparing (1 + x) with the formula, we see that a =

1, b = x, and

...

Does the formula in this case enable us to find the square root of the given expression to any required number of terms?

2. Expand (1+x)-3 to 3 terms.

Here a = 1, b = x, and n = − 3.

..(1+x)=1+(-3)x+

(-3)(-4)2+ (-3)(-4)(-5) 8+...

1.2

1.2.3

This is correct to three decimal places, and by subtracting more terms from 2, the root can be found correct to any required number of places.

Expand the following:

5. (1+2x).

6. (a+3b).

7. (x-3).

8. (a2+b).

9. (2x-3y).

10. (m3+n2).

To expand a trinomial by the Binomial Theorem we put two of the terms equal to a single term, then expand, and finally replace the term substituted by the two terms. Thus, (1+x+x2) may be written (m + x2), and after expansion m may be replaced by 1 + x.

32. Expand to 5 terms (2a+3√ī)→.

33. Expand to 4 terms (V-Vy)8.

Extract the square root of the following by expanding a binomial to four terms, reducing the terms to decimals and