Let’s factorise.

a) $x^2 – 3x + 2$

b) $x^2 – 5x + 6$

c) $x^2 – 6x + 5$

d) $x^2 – 10x + 24$

e) $x^2 – 10x + 21$

f) $x^2 – 8x + 12$

g) $x^2 – 13x + 36$

h) $x^2 – 13x + 30$

Let’s resolve into factors.

a) $x^2 + 3x + 2$

b) $x^2 + 7x + 10$

c) $x^2 + 7x + 12$

d) $x^2 + 8x + 15$

e) $x^2 + 6x + 8$

f) $x^2 + 9x + 20$

g) $x^2 + 9x + 18$

h) $x^2 + 11x + 30$

Let’s represent the following algebraic expressions by drawing algebraic tiles and find their factors.

(a) $x^2 + 3x + 2$

(b) $x^2 + 5x + 4$

(c) $x^2 + 6x + 8$

(d) $x^2 + 7x + 10$

(e) $x^2 + 8x + 12$

(f) $x^2 + 8x + 15$

a) If $a + b = 7$ and $ab = 10$, find the value of $a$ and $b$.

b) If $a + b = -9$ and $ab = 18$, find the value of $a$ and $b$.

c) If $a + b = 4$ and $ab = -32$, find the value of $a$ and $b$.

d) If $a + b = -1$ and $ab = -30$, find the value of $a$ and $b$.

Let’s study carefully, the following tricky ways of factorisation.

$x^2 + 5x + 6 = (x + 2)(x + 3)$

(where $2 + 3 = 5$ and $2 times 3 = 6$)

$x^2 – 5x + 6 = (x – 2)(x – 3)$

(where $-2 – 3 = -5$ and $-2 times (-3) = 6$)

$x^2 + x – 6 = (x – 2)(x + 3)$

(where $-2 + 3 = 1$ and $-2 times 3 = -6$)

$x^2 – x – 6 = (x + 2)(x – 3)$

(where $2 – 3 = -1$ and $2 times (-3) = -6$)

Now, let’s apply the tricky ways of factorization. Say and write the answers as quickly as possible.

a) $x^2 + 3x + 2 =$ ………………..

(where $1 + 2 = 3$ and $1 times 2 = 2$)

b) $x^2 – 3x + 2 =$ ………………..

(where $-1 – 2 = -3$ and $-1 times (-2) = 2$)

c) $x^2 + x – 2 =$ ………………..

(where $2 – 1 = 1$ and $2 times (-1) = -2$)

d) $x^2 – x – 2 =$ ………………..

(where $-2 + 1 = -1$ and $-2 times 1 = -2$)

e) $x^2 + 9x + 20 =$ ………………..

f) $x^2 – 9x + 20 =$ ………………..

Let’s find two numbers whose products and differences are given below.

(a) $text{Product} = 6$, $text{difference} = 1$, required numbers are ……… and ………

(b) $text{Product} = 6$, $text{difference} = 5$, required numbers are ……… and ………

(c) $text{Product} = 10$, $text{difference} = 3$, required numbers are ……… and ………

(d) $text{Product} = 10$, $text{difference} = 9$, required numbers are ……… and ………

(e) $text{Product} = 24$, $text{difference} = 2$, required numbers are ……… and ………

(f) $text{Product} = 24$, $text{difference} = 5$, required numbers are ……… and ………

(g) $text{Product} = 24$, $text{difference} = 10$, required numbers are ……… and ………

(h) $text{Product} = 24$, $text{difference} = 23$, required numbers are ……… and ………

Factorize the given algebraic expressions.

a) $a^2 + 2ab + b^2 – c^2$

b) $x^2 – 2xy + y^2 – z^2$

c) $a^2 + 10a + 25 – b^2$

d) $x^2 + 20x + 100 – y^2$

e) $p^2 – 8p + 16 – q^2$

f) $m^2 – 6mn + 9n^2 – 4p^2$

g) $a^2 – b^2 – c^2 – 2bc$

h) $9x^2 – 16y^2 – 8y – 1$

i) $25w^4 – 9w^2 + 12w – 4$

Let’s find two numbers satisfying the given products and sums.

(a) $text{Product} = 6$, $text{sum} = 5$, required numbers are ……… and ………

(b) $text{Product} = 6$, $text{sum} = 7$, required numbers are ……… and ………

(c) $text{Product} = 8$, $text{sum} = 6$, required numbers are ……… and ………

(d) $text{Product} = 8$, $text{sum} = 9$, required numbers are ……… and ………

(e) $text{Product} = 12$, $text{sum} = -7$, required numbers are ……… and ………

(f) $text{Product} = 12$, $text{sum} = -8$, required numbers are ……… and ………

(g) $text{Product} = 12$, $text{sum} = -13$, required numbers are ……… and ………

Find the factors of the following algebraic expressions.

a) $3x^2 + 12x + 12$

b) $2a^2 – 12a + 18$

c) $8p^2 + 24pq + 18q^2$

d) $25x – 10xy + xy^2$

e) $100m^3 + 60m^2n + 9mn^2$

f) $48b^3d – 120b^2d^2 + 75bd^3$

Resolve into factors:

a) $4a^2 + a + frac{1}{16}$

b) $16x^2 + 4x + frac{1}{4}$

c) $frac{x^2}{9} – 2xy + 9y^2$

d) $frac{b^2}{36} – bc + 9c^2$

e) $frac{a^2}{x^2} + 2 + frac{x^2}{a^2}$

f) $frac{p^2}{q^2} – 2 + frac{q^2}{p^2}$

g) $frac{4m^2}{n^2} – 4 + frac{n^2}{m^2}$

h) $frac{9x^2}{16} – 3a + frac{4a^2}{x^2}$