Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — picking out a single coefficient
Find the coefficient of x3 in the expansion of (2+3x)5.
- 1The general term is (rn)an−rbr with a=2, b=3x, n=5.No need to expand the rest.
- 2For x3, take r=3.The power of b is the power of x wanted.
- 3T=(35)(2)2(3x)3The power of a is n−r=2.
- 4(35)=10, 22=4, (3x)3=27x3The whole of 3x is cubed.
- 510×4×27=1080The coefficient is 1080.
Example 2 — the term independent of x
Find the term independent of x in (x2+x1)6.
- 1General term: (r6)(x2)6−r(x1)r
- 2Collect the powers of x: x2(6−r)×x−r=x12−3rx1=x−1.
- 3Independent of x means the exponent is zero.
- 412−3r=0, so r=4.A whole number, as it must be.
- 5The term is (46)=15.Had r come out fractional, no such term would exist.
Example 3 — an expansion with a fractional power
Expand (1+x)1/2 as far as the term in x3, and use it to estimate 1.02.
- 1(1+x)n=1+nx+2!n(n−1)x2+3!n(n−1)(n−2)x3+⋯ with n=21Valid for ∣x∣<1.
- 2x term: 21x
- 3x2 term: 221×(−21)x2=−8x2The second factor is n−1=−21.
- 4x3 term: 621(−21)(−23)x3=16x3
- 5Put x=0.02: 1+0.01−0.00005=1.00995∣0.02∣<1 ✓
- 6True value 1.0099504… — accurate to seven decimal places.Small x means few terms are needed.