Integer powers
`Find the coefficient of `x^4` in the binomial expansion of `(2+3x)^10
1088640x^4
`Explain how you would use the expansion of `(3-x/5)^10 ` to give an estimate for the value of `2.98^10.
`Substitute `x=0.1` into the expansion
`The coefficient of `x^2` in the binomial expansion of `(1+x/2)^n`, where `n` is a positive integer, is 7. Find the value of `n.
n=8
`Find the binomial expansion of `sqrt(9-x)` up to and including the `x^3` term. State the range of values of x for which the expansion is value.
3+x/6-x^2/216+x^3/3888. abs(x)<9
`Express `(8x+4)/((1-x)(2+x))` as partial fractions.
4/(1-x)-4/(2+x)
`Find the coefficient of `x^3` in the binomial expansion of `(2+x)(3-2x)^7
-24948
`If `x` is so small that terms of `x^3` and higher can be ignored, and `(2-x)(3+x)^4~~a+bx+cx^2`, find the values of the constants `a, b` and `c.
a=162, b=135, c=0
g(x)=(4+kx)^5`, where `k` is a constant. Given that the coefficient of `x^3` in the binomial expansion of g(`x`) is 20, find the value of `k.
k=1/2
`Find the binomial expansion of `5/(3+2x)` up to and including the `x^3` term. State the range of values of x for which the expansion is value.
5/3-10/9x+20/27x^2-40/81x^3. abs(x)<3/2
g(x)=(12x-1)/((1+2x)(1-3x)), abs(x)<1/3`. Given that `g(x)` can be expressed in the form `g(x)=A/(1+2x)+B/(1-3x)`, find the values of `A` and `B.
A=-14/5, B=9/5
`Find the binomial expansion of `(x+1/x)^5` giving each term in its simplest form.
x^5+5x^3+10x+10/x+5/x^3+1/x^5
`Using the first four terms of the binomial expansion `(2+x/5)^10`, and by substituting an appropriate value for `x`, find an appropriate value for `2.1^10.
1666.56
`The first 3 terms in the binomial expansion of `(2+px)^7`, where `p` is a constant, are 128, 2240`x` and `qx^2`, where `q` is a constant. Find the value of `p` and the value of `q.
p=5, q=16800
m(x)=sqrt(4-x), abs(x)<4. `Find the exact value of `m(x)` when `x=1/9.
sqrt(35)/3
(3x^2+4x-5)/((x+3)(x-2)) = A + B/(x+3)+C/(x-2). `Find the values of the constants `A, B` and `C.
A=3, B=-2` and `C=3
`The coefficient of `x^3` in the expansion of `(2+x)(3-ax)^4` is 30. Find three possible values of the constant `a.
1, (1+-sqrt(105))/8
`Using the first four terms of the binomial expansion of `(1-5x)^30`, estimate the value of `0.995^30`, giving your answer to 6 decimal places.
0.860368
`The coefficient of `x^2` in the binomial expansion of `(2+kx)^8`, where `k` is a positive constant, is 2800. Use algebra to find the value of `k.
k=1.25
`The first 3 terms in the binomial expansion of `1/sqrt(a+bx)` are `3+1/3x+1/18x^2+... `Find the values of `a` and `b.
a=1/9, b=-2/81
(1+x)/(1+3x)~~1-2x+6x^2-18x^3`. Taking a suitable value of `x`, which should be stated, use the series expansion above, to find an approximate value for `101/103`, giving your answer to 5 decimal places.
x=0.01, 0.98058
`Find the coefficient of `x^4` in the binomial expansion of `(3-2x^2)^9
314928
`By expanding `(1+2x)^12` in ascending powers of `x` up to and including `x^3`, calculate an approximate value of `1.02^12
1.26816, x=0.01
`Find the coefficient of `x^2` in the expansion of `(1+2x)^8(2-5x)^7
-4704
`When `(1+ax)^n` is expanded as a series in ascending powers of `x`, the coefficients of `x` and `x^2` are -6 and 27 respectively. Find the values of `a` and `n.
n=-2, a=3
`Without using partial fractions, obtain the first four non-zero terms in the expansion, in ascending powers of `x`, of the function `f(x)` where `f(x) = 1/sqrt(1+3x^2), 3x^2<1
1-(3x^2)/2+(27x^4)/8-(135x^6)/16