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

Let’s factorise.

a) $4a^2 + 4a + 1$

b) $4a^2 + 20a + 25$

c) $9a^2 + 6a + 1$

d) $9x^2 + 12x + 4$

e) $9a^2 + 24a + 16$

f) $16x^2 + 24x + 9$

g) $9x^2 – 24x + 16$

h) $25x^2 – 80x + 64$

i) $16a^2 – 40a + 25$

j) $49p^2 – 14p + 1$

k) $36 – 60p + 25p^2$

l) $144 – 120x + 25x^2$

Let’s resolve into factors.

a) $x^2 + 6x + 9$

b) $x^2 + 8x + 16$

c) $x^2 + 10x + 25$

d) $a^2 + 12a + 36$

e) $a^2 + 14a + 49$

f) $a^2 + 16a + 64$

g) $x^2 – 4x + 4$

h) $x^2 – 2x + 1$

i) $p^2 – 18p + 81$

j) $p^2 – 20p + 100$

k) $p^2 – 24p + 144$

l) $x^2 – 30x + 225$

Let’s find the expressions to be inserted to make the following expressions perfect squares.

a) $x^2 +$ ……. $+ 4$

b) $x^2 +$ ……. $+ 9$

c) $a^2 -$ ……. $+ 16$

d) $a^2 -$ ……. $+ 64$

e) $9x^2 +$ ……. $+ 4y^2$

f) $25x^2 -$ ……. $+ 81y^2$

Let’s rewrite these expressions in the forms of $a^2 + 2ab + b^2$ or $a^2 – 2ab + b^2$.

E.g.

$x^2 + 2x + 1 = x^2 + 2.x.1 + 1^2$

$9x^2 – 30x + 25 = (3x)^2 – 2.3x.5 + 5^2$

a) $x^2 + 4x + 4 =$ ………………..

b) $x^2 – 8x + 16 =$ ………………..

c) $4x^2 + 12x + 9 =$ ………………..

d) $9x^2 – 12x + 4 =$ ………………..

Let’s express the following algebraic expressions as perfect squares.

a) $x^2 + 10x + 25$

b) $a^2 + 20a + 100$

c) $4x^2 + 28x + 49$

d) $4x^2 + 12xy + 9y^2$

e) $a^2 – 6a + 9$

f) $x^2 – 16x + 64$

g) $9m^2 – 12mn + 4n^2$

h) $25a^2 – 20ab + 4b^2$

Simplify.

a) $frac{2.1 times 2.1 – 0.9 times 0.9}{2.1 – 0.9}$

b) $frac{3.6 times 3.6 – 1.4 times 1.4}{3.6 + 1.4}$

c) $frac{2.5 times 2.5 – 1.4 times 1.4}{2.5 + 1.4}$

d) $frac{5.6 times 5.6 – 4.4 times 4.4}{5.6 – 4.4}$

Let’s say and write the following algebraic expressions as the perfect square forms.

E.g.

$a^2 + 2.a.b + b^2 = (a + b)^2$

$(3x)^2 – 2.3x.5 + 5^2 = (3x – 5)^2$  

Let’s simplify by factorisation process.

a) $25^2 – 15^2$

b) $64^2 – 44^2$

c) $96^2 – 86^2$

d) $(101)^2 – (100)^2$

e) $51 times 49$

f) $82 times 78$

g) $101 times 99$

h) $103 times 97$