Expand and simplify $$(3-x)^3$$
$27 - 27x + 9x^2 - x^3$
Expand and simplify $$(5+2x)^4$$
$625 + 1000x + 600x^2 + 160x^3 + 16x^4$
Expand and simplify $$(3-4x)^5$$
$243 - 1620x + 4320x^2 - 5760x^3 + 3840x^4 - 1024x^5$
Expand and simplify $$\left(3+\frac{1}{2}x\right)^4$$
$81 + 54x + \dfrac{27}{2}x^2 + \dfrac{3}{2}x^3 + \dfrac{1}{16}x^4$
Find the first four terms in ascending powers of $x$ in the expansion of $$(1-x)^6$$
$1 - 6x + 15x^2 - 20x^3$
Find the first four terms in ascending powers of $x$ in the expansion of $$(1-\frac{1}{2}x)^7$$
$1 - \dfrac{7}{2}x + \dfrac{21}{4}x^2 - \dfrac{35}{8}x^3$
Find the first four terms in ascending powers of $x$ in the expansion of $$(2+x)^9$$
$512 + 2304x + 4608x^2 + 5376x^3$
Find the first four terms in ascending powers of $x$ in the expansion of $$(2+5x)^{10}$$
$1024 + 25600x + 288000x^2 + 1920000x^3$
Find the coefficient of $x^4$ in the expansion of $$(1-x)^{14}$$
$1001$
Find the coefficient of $x^4$ in the expansion of $$(1-\frac{1}{3}x)^{12}$$
$\dfrac{55}{9}$
Expand and simplify $$(1-\sqrt{3})^4$$
$28 - 16\sqrt{3}$
Expand and simplify $$(1+2\sqrt{3})^4$$
$217 + 104\sqrt{3}$
- Expand and simplify $(1+x)^6$ in ascending powers of $x$ up to and including the term in $x^3$
- By substituting a suitable value for $x$, find an estimate for $1.02^6$ to 4 decimal places
- By substituting a suitable value for $x$, find an estimate for $0.99^6$ to 4 decimal places
- Find the percentage errors in these estimates to 3 significant figures
- Explain why the estimate for $0.99^6$ has a smaller error
- $1 + 6x + 15x^2 + 20x^3$
- $1.1262$
- $0.9415$
- $0.00334\%$ and $0.00211\%$
- The value of $x$ used for this estimate is closer to 0
- Expand and simplify $(1+2x)^8$ in ascending powers of $x$ up to and including the term in $x^3$
- By substituting a suitable value for $x$, find an estimate for $1.01^8$ to 4 decimal places
- Find the percentage error in this estimate to 3 significant figures
- $1 + 16x + 112x^2 + 448x^3$
- $1.0829$
- $0.00400\%$
Find the term independent of $x$ in the expansion of $$\left(2x - \dfrac{1}{2x}\right)^{12}$$
$924$
Find the term independent of $x$ in the expansion of $$\left(\dfrac{1}{x} + x^2\right)^6$$
$15$
Expand and simplify $$(1 + \sqrt{5})^3 - (1 - \sqrt{5})^3$$
$16\sqrt{5}$
Expand and simplify $$(1 + 10x)^4 + (1 - 10x)^4$$
$2 + 1200x^2 + 20000x^4$
Expand and simplify $$(1 + 4x)(1 + x)^3$$
$1 + 7x + 15x^2 + 13x^3 + 4x^4$
Expand and simplify $$(1 - x)\left(1 + \dfrac{1}{x}\right)^3$$
$-x - 2 + \dfrac{2}{x^2} + \dfrac{1}{x^3}$
Expand and simplify $$(1 + x)(1 - 3x)^{10}$$ up to and including the term in $x^3$.
$1 - 29x + 375x^2 - 2835x^3$
Expand and simplify $$(1 - 2x)(1 + x)^8$$ up to and including the term in $x^3$.
$1 + 6x + 12x^2$
Expand and simplify $$(1 + x + x^2)(1 - x)^6$$ up to and including the term in $x^3$.
$1 - 5x + 10x^2 - 11x^3$
Expand and simplify $$(1 + 3x - x^2)(1 + 2x)^7$$ up to and including the term in $x^3$.
$1 + 17x + 125x^2 + 518x^3$
Expand $$(1 - x)^5$$ in ascending powers of $x$.
$1 - 5x + 10x^2 - 10x^3 + 5x^4 - x^5$
By first expanding $(\sqrt{3} + 1)(\sqrt{3} - 2)$, expand and simplify $$(\sqrt{3} + 1)^5(\sqrt{3} - 2)^5$$
$76 - 44\sqrt{3}$
The term independent of $x$ in the expansion of $$\left(x^3 + \dfrac{a}{x^2}\right)^5$$ is $-80$. Find $a$.
$-2$
Find the first 3 terms in ascending powers of $x$ in the expansion of $$\left(1 + \dfrac{x}{2}\right)^8(1 - x)^6$$
$1 - 2x - 2x^2$
The first two terms in ascending powers of $x$ of the expansion of $$\left(1 + \dfrac{ax}{2}\right)^{10} + (1 + bx)^{10}$$ are $2$ and $90x^2$. Given $a < b$, find $a$ and $b$.
$a = -2$ and $b = 1$
Find the coefficient of $x^2$ in the expansion of $$(1 + 2x)^2(1 + 4x)^7$$
$452$
Given $$(k - x)^9 = a - bx + bx^2 + \ldots$$ find positive integers $a$, $b$, and $k$.
$a = 262144$, $b = 589824$, and $k = 4$
The coefficient of $x^2$ in the binomial expansion of $$\left(1 + \dfrac{2}{5}x\right)^n$$ is $1.6$. Find the coefficient of $x^4$.
$n = 5$
In the expansion of $$\left(1 + \dfrac{x}{k}\right)^n$$ where $k$ is a constant, the coefficient of $x^2$ is three times the coefficient of $x^3$, and the coefficient of $x$ is $2$. Find $k$.
$k = 2$
In the expansion of $$(1 + px)^q$$ the coefficient of $x$ is $-12$ and the coefficient of $x^2$ is $60$. Find the coefficient of $x^3$.
$-160$
In the expansion of $$\left(1+\dfrac{x}{k}\right)^{2n}$$ the coefficient of $x^3$ is half the coefficient of $x^2$ and the coefficient of $x$ is $2$. Find $n$ and $k$.
$n = k = 4$
In the expansion of $$(1-x)(1+2x)^{n}$$ the coefficient of $x^2$ is $198$. Find the coefficient of $x^3$.
$1100$