Chapter 1: Algebra (Expansion and Factorisation)
Extension Problems
Simplify $(x-y+z)(x+y-z)-(x+y+z)(x-y-z)$
*hint: expanding everything would take a long time. Can you replace two variables with one variable?
If $a+b=2$ and $ab=-1$, evaluate $(a^2-1)(b^2-1)$.
If $\frac{y}{x}+\frac{x}{y}=3$, evaluate $\frac{2x^2-3xy+2y^2}{x^2+y^2}$.
Evaluate $\frac{2003^2-2001\times2003-2}{2}$ without using a calculator.
If $5x=4y$, evaluate $\frac{x^2+xy+y^2}{x^2-xy+y^2}$.
*Give your answer as an improper fraction in simplest terms.
If $a+b=5$, $c+d=2$ and $ad=bc=1$, evaluate:
a) $ac+bd$
b) $\frac{a}{c}$. Give your answer as an improper fraction in simplest terms.
Factorization Extension Level 1
Fully factorize.
$x^2+x+\frac{1}{4}$
$x^2-\frac{2}{3}x+\frac{1}{9}$
$\left(a-b\right)x-\left(b-a\right)y$
$3\left(x+y\right)^3+27\left(x+y\right)^2$
$\left(a-b\right)^2-9b^2$
$\left(a-1\right)x^2+4\left(1-a\right)y^2$
$\left(3a+2b\right)^2-\left(-4a+b\right)^2$
$\left(a^2+b^2\right)^2-4a^2b^2$
Factorization Extension Level 2
Fully factorize.
$ax^2-\left(a+2\right)x+2$
$x^2-\left(a+1\right)x+a$
$ax^2+\left(2a-1\right)x-2$
$abx^2+\left(a+b\right)x+1$
$ax^2-\left(1+ab\right)x+b$
$abx^2+\left(2a^2-b^2\right)x-2ab$
Factorization Extension Level 3
Fully factorize.
$\left(x+y\right)^2+\left(x+y\right)-2$
$\left(a-b\right)^2-3\left(a-b\right)-10$
$\left(x+y\right)^2-4\left(x-y\right)^2$
$\left(x^2+x\right)^2+3\left(x^2+x\right)-10$
Factorization Extension Level 4
Fully factorize.
$a^2\left(x^2-a^2\right)-b^2\left(x^2-b^2\right)$
$2x^2-xy-y^2-7x+y+6$
$2x^2+xy-6y^2-4x-y+2$
$x-y-x^2+2xy-y^2+2$
$a^2b-2abc-b-ab^2+a-2c$
$x^3-\left(b-1\right)x^2+abx-ab\left(b-1\right)$
$a^2-ab+ac-2bc-2c^2$
$x^3+\left(a+2\right)x^2+\left(2a+1\right)x+a$
$a^2b-ab^2-a^2c-ac^2-b^2c+bc^2+2abc$