Question 1
Expand
a) $(x+y)^3$
b) $(x-y)^3$
c) $(x+y)(x^2-xy+y^2)$
d) $(x-y)(x^2+xy+y^2)$
Question 2
Expand
a) $(x+1)^3$
b) $(x+2)^3$
c) $(x+5)^3$
d) $(7+x)^3$
e) $(2x+1)^3$
f) $(3x+2)^3$
g) $(6y+z)^3$
h) $(a^2+2b)^3$
Question 3
Expand
a) $(x-1)^3$
b) $(x-2)^3$
c) $(2x-1)^3$
d) $(2a-b)^3$
e) $(4x-3y)^3$
f) $(x^2-5)^3$
g) $(x^2-x)^3$
h) $(2a^2-3b^3)^3$
Question 4
Factorise
a) $x^3+y^3$
b) $x^3+1$
c) $a^3+8$
d) $m^3+125$
e) $8p^3+1$
f) $27z^3+64$
Question 5
Factorise
a) $x^3-y^3$
b) $x^3-1$
c) $y^3-64$
d) $27g^3-1$
e) $125g^3-27$
f) $343a^3-8p^3$
Question 6
Expand and simplify
a) $\left(x+y\right)^3+\left(x-y\right)^3$
b) $\left(x+y\right)^3-\left(x-y\right)^3$
c) $\left(x-y\right)^3-\left(x+y\right)^3$
d) $\left(2x^2-3y\right)^3+\left(2x^2+3y\right)^3$
Question 7
Factorise fully
a) $2x^3+2y^3$
b) $x^4-xy^3$
c) $x^3+y^3+x+y$
d) $2a^3+6a^2b+6ab^2+2b^3$
e) $x^3-y^3+x^2-y^2$
f) $a^6+2a^3b^3+b^6$
Question 8
Simplify
a) $\frac{x^3+y^3}{x+y}$
b) $\frac{5x^3-5y^3}{x-y}$
c) $\frac{x^3-y^3}{6x-6y}$
d) $\frac{125t^3-1}{25t^2-1}$
e) $\frac{8a^3+b^3}{6a^2+ab-b^2}$
f) $\frac{x^3-y^3}{x-y}\div\frac{x^3+y^3+2xy(x+y)}{x+y}$
Question 9
Factorise fully
a) $x^6+1$
b) $x^6-1$
c) $x^{24}+y^{24}$
d) $x^8-x^2$
Question 10
Answer the following
a) If $x$ and $y$ are real numbers such that $x+y=7$ and $x^3+y^3=91$, find the value of $xy$.
b) If $x$ and $y$ are real numbers such that $x-y=1$ and $x^3-y^3=61$, find the value of $xy$.
Question 11
Answer the following
a) Evaluate $\sqrt{\sqrt[3]{1000}+\sqrt[3]{1000+150(15)+125}}$
Answers
Question 1
a) $x^3+3x^2y+3xy^2+y^3$
b) $x^3-3x^2y+3xy^2-y^3$
c) $x^3+y^3$
d) $x^3-y^3$
Question 2
a) $x^3+3x^2+3x+1$
b) $x^3+6x^2+12x+8$
c) $x^3+15x^2+75x+125$
d) $343+147x+21x^2+x^3$
or
$x^3+21x^2+147x+343$
e) $8x^3+12x^2+6x+1$
f) $27x^3+54x^2+36x+8$
g) $216y^3+108y^2z+18yz^2+z^3$
h) $a^6+6a^4b+12a^2b^2+8b^3$
Question 3
a) $x^3-3x^2+3x-1$
b) $x^3-6x^2+12x-8$
c) $8x^3-12x^2+6x-1$
d) $8a^3-12a^2b+6ab^2-b^3$
e) $64x^3-144x^2y+108xy^2-27y^3$
f) $x^6-15x^4+75x^2-125$
g) $x^6-3x^5+3x^4-x^3$
h) $8a^6-36a^4b^3+54a^2b^6-27b^9$
Question 4
a) $\left(x+y\right)\left(x^2-xy+y^2\right)$
b) $\left(x+1\right)\left(x^2-x+1\right)$
c) $\left(a+2\right)\left(a^2-2a+4\right)$
d) $\left(m+5\right)\left(m^2-5m+25\right)$
e) $\left(2p+1\right)\left(4p^2-2p+1\right)$
f) $\left(3z+4\right)\left(9z^2-12z+16\right)$
Question 5
a) $\left(x-y\right)\left(x^2+xy+y^2\right)$
b) $\left(x-1\right)\left(x^2+x+1\right)$
c) $\left(y-4\right)\left(y^2+4y+16\right)$
d) $\left(3g-1\right)\left(9g^2+3g+1\right)$
e) $\left(5g-3\right)\left(25g^2+15g+9\right)$
f) $\left(7a-2p\right)\left(49a^2+14ap+4p^2\right)$
Question 6
a) $2x^3+6xy^2$
b) $6x^2y+2y^3$
c) $-6x^2y-2y^3$
d) $16x^6+108x^2y^2$
Question 7
a) $2\left(x+y\right)\left(x^2-xy+y^2\right)$
b) $x\left(x-y\right)\left(x^2+xy+y^2\right)$
c) $(x + y) (x^2 - x y + y^2 + 1)$
d) $2(a+b)^3$
e) $(x - y) (x + x^2 + y + x y + y^2)$
f) $(a + b)^2(a^2 - a b + b^2)^2$
Question 8
a) $x^2 - x y + y^2$
b) $5 (x^2 + x y + y^2)$
c) $\tfrac16\left(x^2+xy+y^2\right)$
d) $\frac{\left(25t^2+5t+1\right)}{5t+1}$
e) $\frac{\left(4a^2-2ab+b^2\right)}{3a-b}$
f) $1$
Question 9
a) $\left(x^2+1\right)\left(x^4-x^2+1\right)$
b) $\left(x+1\right)\left(x-1\right)\left(x^2-x+1\right)\left(x^2+x+1\right)$
c) $(x^8 + y^8) (x^{16} - x^8 y^8 + y^{16})$
d) $x^2\left(x+1\right)\left(x-1\right)\left(x^2-x+1\right)\left(x^2+x+1\right)$
Question 10
a) $xy=12$
b) $xy=20$
Question 11
a) $\sqrt{\sqrt[3]{1000}+\sqrt[3]{1000+150(15)+125}}=5$
Question 1
Expand
a) $(x+y)^3$
b) $(x-y)^3$
c) $(x+y)(x^2-xy+y^2)$
d) $(x-y)(x^2+xy+y^2)$
Question 2
Expand
a) $(x+1)^3$
b) $(x+2)^3$
c) $(x+5)^3$
d) $(7+x)^3$
e) $(2x+1)^3$
f) $(3x+2)^3$
g) $(6y+z)^3$
h) $(a^2+2b)^3$
Question 3
Expand
a) $(x-1)^3$
b) $(x-2)^3$
c) $(2x-1)^3$
d) $(2a-b)^3$
e) $(4x-3y)^3$
f) $(x^2-5)^3$
g) $(x^2-x)^3$
h) $(2a^2-3b^3)^3$
Question 4
Factorise
a) $x^3+y^3$
b) $x^3+1$
c) $a^3+8$
d) $m^3+125$
e) $8p^3+1$
f) $27z^3+64$
Question 5
Factorise
a) $x^3-y^3$
b) $x^3-1$
c) $y^3-64$
d) $27g^3-1$
e) $125g^3-27$
f) $343a^3-8p^3$
Question 6
Expand and simplify
a) $\left(x+y\right)^3+\left(x-y\right)^3$
b) $\left(x+y\right)^3-\left(x-y\right)^3$
c) $\left(x-y\right)^3-\left(x+y\right)^3$
d) $\left(2x^2-3y\right)^3+\left(2x^2+3y\right)^3$
Question 7
Factorise fully
a) $2x^3+2y^3$
b) $x^4-xy^3$
c) $x^3+y^3+x+y$
d) $2a^3+6a^2b+6ab^2+2b^3$
e) $x^3-y^3+x^2-y^2$
f) $a^6+2a^3b^3+b^6$
Question 8
Simplify
a) $\frac{x^3+y^3}{x+y}$
b) $\frac{5x^3-5y^3}{x-y}$
c) $\frac{x^3-y^3}{6x-6y}$
d) $\frac{125t^3-1}{25t^2-1}$
e) $\frac{8a^3+b^3}{6a^2+ab-b^2}$
f) $\frac{x^3-y^3}{x-y}\div\frac{x^3+y^3+2xy(x+y)}{x+y}$
Question 9
Factorise fully
a) $x^6+1$
b) $x^6-1$
c) $x^{24}+y^{24}$
d) $x^8-x^2$
Question 10
Answer the following
a) If $x$ and $y$ are real numbers such that $x+y=7$ and $x^3+y^3=91$, find the value of $xy$.
b) If $x$ and $y$ are real numbers such that $x-y=1$ and $x^3-y^3=61$, find the value of $xy$.
Question 11
Answer the following
a) Evaluate $\sqrt{\sqrt[3]{1000}+\sqrt[3]{1000+150(15)+125}}$
Answers
Question 1
a) $x^3+3x^2y+3xy^2+y^3$
b) $x^3-3x^2y+3xy^2-y^3$
c) $x^3+y^3$
d) $x^3-y^3$
Question 2
a) $x^3+3x^2+3x+1$
b) $x^3+6x^2+12x+8$
c) $x^3+15x^2+75x+125$
d) $343+147x+21x^2+x^3$
or
$x^3+21x^2+147x+343$
e) $8x^3+12x^2+6x+1$
f) $27x^3+54x^2+36x+8$
g) $216y^3+108y^2z+18yz^2+z^3$
h) $a^6+6a^4b+12a^2b^2+8b^3$
Question 3
a) $x^3-3x^2+3x-1$
b) $x^3-6x^2+12x-8$
c) $8x^3-12x^2+6x-1$
d) $8a^3-12a^2b+6ab^2-b^3$
e) $64x^3-144x^2y+108xy^2-27y^3$
f) $x^6-15x^4+75x^2-125$
g) $x^6-3x^5+3x^4-x^3$
h) $8a^6-36a^4b^3+54a^2b^6-27b^9$
Question 4
a) $\left(x+y\right)\left(x^2-xy+y^2\right)$
b) $\left(x+1\right)\left(x^2-x+1\right)$
c) $\left(a+2\right)\left(a^2-2a+4\right)$
d) $\left(m+5\right)\left(m^2-5m+25\right)$
e) $\left(2p+1\right)\left(4p^2-2p+1\right)$
f) $\left(3z+4\right)\left(9z^2-12z+16\right)$
Question 5
a) $\left(x-y\right)\left(x^2+xy+y^2\right)$
b) $\left(x-1\right)\left(x^2+x+1\right)$
c) $\left(y-4\right)\left(y^2+4y+16\right)$
d) $\left(3g-1\right)\left(9g^2+3g+1\right)$
e) $\left(5g-3\right)\left(25g^2+15g+9\right)$
f) $\left(7a-2p\right)\left(49a^2+14ap+4p^2\right)$
Question 6
a) $2x^3+6xy^2$
b) $6x^2y+2y^3$
c) $-6x^2y-2y^3$
d) $16x^6+108x^2y^2$
Question 7
a) $2\left(x+y\right)\left(x^2-xy+y^2\right)$
b) $x\left(x-y\right)\left(x^2+xy+y^2\right)$
c) $(x + y) (x^2 - x y + y^2 + 1)$
d) $2(a+b)^3$
e) $(x - y) (x + x^2 + y + x y + y^2)$
f) $(a + b)^2(a^2 - a b + b^2)^2$
Question 8
a) $x^2 - x y + y^2$
b) $5 (x^2 + x y + y^2)$
c) $\tfrac16\left(x^2+xy+y^2\right)$
d) $\frac{\left(25t^2+5t+1\right)}{5t+1}$
e) $\frac{\left(4a^2-2ab+b^2\right)}{3a-b}$
f) $1$
Question 9
a) $\left(x^2+1\right)\left(x^4-x^2+1\right)$
b) $\left(x+1\right)\left(x-1\right)\left(x^2-x+1\right)\left(x^2+x+1\right)$
c) $(x^8 + y^8) (x^{16} - x^8 y^8 + y^{16})$
d) $x^2\left(x+1\right)\left(x-1\right)\left(x^2-x+1\right)\left(x^2+x+1\right)$
Question 10
a) $xy=12$
b) $xy=20$
Question 11
a) $\sqrt{\sqrt[3]{1000}+\sqrt[3]{1000+150(15)+125}}=5$
