Factorize : 

a) $x(a + b) + y(a + b) =$ ………………..

b) $a(p + q) – (p + q) =$ ………………..

c) $(y – 2) – y(y – 2) =$ ………………..

d) $x(a – b) + 2(a – b) =$ ………………..

a) If $a + frac{1}{a} = 3$, find the values of (i) $a^2 + frac{1}{a^2}$ (ii) $left(a – frac{1}{a}right)^2$

b) If $x + frac{1}{x} = 5$, find the values of (i) $x^2 + frac{1}{x^2}$ (ii) $left(x – frac{1}{x}right)^2$

c) If $p – frac{1}{p} = 4$, find the values of (i) $p^2 + frac{1}{p^2}$ (ii) $left(p + frac{1}{p}right)^2$

Let’s factorise, say and write the answers as quickly as possible.

a) $ax + bx =$ ………………..

b) $ax – ay =$ ………………..

c) $y^2 – y =$ ………………..

d) $2x^3 + x^2 =$ ………………..

e) $x^2 – x^3 =$ ………………..

f) $x^2y + xy^2 =$ ………………..

a) If $(x + y) = 5$ and $xy = 3$, find the value of $x^2 + y^2$.

b) If $(a – b) = 4$ and $ab = 2$, find the value of $a^2 + b^2$.

c) If $left(x + frac{1}{x}right) = 3$, find the value of $x^2 + frac{1}{x^2}$.

d) If $left(p – frac{1}{p}right) = 7$, find the value of $p^2 + frac{1}{p^2}$.

Remember:

$(x + y)^2 = x^2 + 2xy + y^2$

and, $x^2 + y^2 = (x + y)^2 – 2xy$

Let’s find the products using the formula $(a + b)(a – b) = a^2 – b^2$.

Hint: $99 times 101 = (100 – 1) times (100 + 1) = 100^2 – 1^2 = 10000 – 1 = 9999$

a) $19 times 21$

b) $49 times 51$

c) $78 times 82$

d) $102 times 98$

Let’s apply the appropriate formula to find the products.

a) $(a + 3)(a – 3)$

b) $(2x + 1)(2x – 1)$

c) $(4 + 3p)(4 – 3p)$

d) $(2x – 3y)(2x + 3y)$

e) $(x^2 – y^2)(x^2 + y^2)$

f) $(2x^2 + 5y^2)(2x^2 – 5y^2)$

g) $(a + b + c)(a + b – c)$

h) $(x – y + z)(x + y + z)$

i) $(p – q – r)(p + q – r)$

j) $(x + y)(x – y)(x^2 + y^2)$

k) $(x + 2)(x – 2)(x^2 + 4)$

l) $(2a + y)(2a – y)(4a^2 + y^2)$

a) Find the squares of

(i) $2a + 1$

(ii) $x – 3y$

(iii) $a – frac{1}{a}$

(iv) $x + frac{1}{x}$

b) Expand

(i) $(3x – 1)^2$

(ii) $left(2y + frac{1}{2y}right)^2$

Let’s say and write the expanded forms of these expressions.

Let’s say and write the following products in the forms of difference of two squared terms.

a) $(x + y)(x – y) =$ ………………..

b) $(x + 1)(x – 1) =$ ………………..

c) $(p + 2)(p – 2) =$ ………………..

d) $(m + 3)(m – 3) =$ ………………..

e) $left(a + frac{1}{a}right)left(a – frac{1}{a}right) =$ ………………..

f) $left(x + frac{2}{x}right)left(x – frac{2}{x}right) =$ ………………..

g) $left(frac{3}{p} + pright)left(frac{3}{p} – pright) =$ ………………..

h) $left(frac{4}{m} + mright)left(frac{4}{m} – mright) =$ ………………..

Write the degrees of the following polynomials.

(i) $3x^2$

(ii) $2a^2bc$

(iii) $2x^3 – 4x^2 + 7x$

(iv) $x^3y^2 + x^2y^2 – xy$