Algebraic Expressions-I (Polynomials and Factorization)

Let’s say and write the polynomial expressions in the table.
a) $2x + 3$

b) $3y^2 – 2y + 4$

c) $a + frac{1}{a}$

d) $sqrt{5x} + 1$

e) $frac{x^2}{2} – 3x + 7$

f) $sqrt{x + 2} – 3$

Let’s say and write the degrees of these polynomials.

a) $3x$ …………………

b) $2p^2$ …………………

c) $4a^2b$ …………………

d) $7y – 1$ …………………

e) $2x^3 – 3x^2 + 6$ …………………

f) $9xy – 7x + 2y$ …………………

Let’s say and write the values of the expressions as quickly as possible.

a) If $l = 4$ and $b = 3$, then (i) $l times b =$ …….. (ii) $2(l + b) =$ ……..

b) If $l = 5$, then (i) $l^2 =$ …….. (ii) $l^3 =$ …….. (iii) $6l^2 =$ ……..

Let’s answer the following questions.

a) Why is $frac{x^2}{2}$ a polynomial but $frac{2}{x^2}$ is not a polynomial?

b) Why is $sqrt{2x}$ not a polynomial but $sqrt{2}x$ is a polynomial?

c) Define the degree of a polynomial with examples.

Find the degrees of the following polynomials.

a) $2xyz$

b) $5a^2bc$

c) $3y^2 – 4y + 8$

d) $7x^5 – 2x^4 + 5x^3 – 7x^2 – 3x$

e) $x^2 + y^2 + z^2 – 3xyz$

f) $x^3y^3 – 2x^2y^2 + 7xy – 5$

Which of the following expressions are polynomials? Write with reason.

(i) $x^2 + frac{1}{2}$

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

(iii) $sqrt{3}x – 2$

(iv) $2sqrt{x} + 5$

Write the degrees of the following polynomials.

(i) $3x^2$

(ii) $2a^2bc$

(iii) $2x^3 – 4x^2 + 7x$

(iv) $x^3y^2 + x^2y^2 – xy$

Find the squares of: a) $3x + 2y$ b) $left(x – frac{1}{x}right)$

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

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 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) $(2a + y)(2a – y)(4a^2 + y^2)$

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) $102 times 98$

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, $x^2 + y^2 = (x + y)^2 – 2xy$

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) $x^2y + xy^2 =$ ………………..

Factorize : 

a) $x(a + b) + y(a + b) =$ ………………..

b) $a(p + q) – (p + q) =$ ………………..

c) $(y – 2) – y(y – 2) =$ ………………..

d) $x(a – b) + 2(a – b) =$ ………………..

Let’s factorise

a) $2ax + 4ay$

b) $6x^2 – 9x$

c) $8px^3 + 12px^2$

d) $4mx^2 – 6mx + 2m^2x$

e) $9b^2y^3 – 6b^3y^2 + 15b^2y^2$

f) $-9a^5 – 6a^3x^2 – 15a^2$

g) $x(2x + 3) + 4(2x + 3)$

h) $2y(y – 4) – 3(y – 4)$

i) $7x(x + 2y) + y(x + 2y)$

Let’s apply the process of factorisation and simplify.

a) $15 times 6 + 15 times 4$

b) $3.4 times 10^3 – 2.6 times 10^3$

c) $3.5 times 10^{-2} + 2.3 times 10^{-2}$

Let’s resolve into factors.

a) $ax + ay + bx + by$

b) $px + qx – py – qy$

c) $x^2 + 2x + 3x + 6$

d) $x^2 – 4x – 3x + 12$

e) $6x^2 – 2y^2 + 4xy – 3xy$

f) $ax^2 + ay^2 + bx^2 + by^2$

g) $x^3 + x^2 + x + x^2y + xy + y$

h) $x^2 – x(y + z) + yz$

i) $a^2 – a(2x – y) – 2xy$

j) $x^2 – (y – 3)x – 3y$

k) $x(a^2 – b^2) + a(b^2 – x^2)$

l) $ab(c^2 + 1) – c(a^2 + b^2)$

Let’s find the length and breadth of the following rectangles of the given area.

a) Area $= 3a^2b + 6ab$

b) Area $= x^2 + 2x + xy + 2y$

c) Area $= x^2 + 3x + 4ax + 12a$

a) If $(4ax + 6ay)$ sq. unit is the area of a rectangle, find the length and breadth of the rectangle.

b) Find the length and breadth of a rectangle of area $(2x^2 + 2xy + xy + y^2)$ sq. unit. Also, find the perimeter of the rectangle.

c) The area of a rectangular field is $(x^2 + 3xy + 2xy + 6y^2)$ sq. unit. Find the perimeter of the field.

Let’s say and write the following expressions as the product of their factors.

Let’s say and write the differences of these squared numbers as quickly as possible.

a) $10^2 – 9^2 =$ ………………..

b) $7^2 – 5^2 =$ ………………..

c) $21^2 – 20^2 =$ ………………..

d) $50^2 – 49^2 =$ ………………..

e) $20^2 – 10^2 =$ ………………..

f) $40^2 – 30^2 =$ ………………..

Let’s resolve into factors.

a) $x^2 – 36$

b) $a^2 – 49$

c) $25 – y^2$

d) $16 – 81p^2$

e) $x^3 – 16x$

f) $2a^2 – 72$

g) $5p^3 – 80p$

h) $3y^3 – 27y$

i) $5a^3 – 20ab^2$

j) $8x^5y – 18x^3y^3$

k) $25x^2 – frac{1}{49}$

l) $frac{1}{4x^2} – frac{1}{81}$

Let’s factorise:

a) $x^4 – 16$

b) $a^4 – 81$

c) $x^4 – y^4$

d) $16p^4 – q^4$

e) $81x^4 – 625$

f) $32y^4 – 162$

g) $a^8 – b^8$

h) $256x^8 – y^8$

Let’s factorise:

a) $(a – b)^2 – 9$

b) $(x + y)^2 – 25$

c) $(a^2 + b^2)^2 – 4$

d) $(a^2 – b^2)^2 – a^2b^2$

e) $16 – (x + y)^2$

f) $49 – (a – b)^2$

g) $(x + 3)^2 – (y + 2)^2$

h) $(a – 5)^2 – (b – 4)^2$

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) $103 times 97$

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 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) $25a^2 – 20ab + 4b^2$

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 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 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) $144 – 120x + 25x^2$

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

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) $48b^3d – 120b^2d^2 + 75bd^3$

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 ………

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 ………

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 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 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 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) $x^2 + x – 2$

b) $x^2 + 2x – 3$

c) $x^2 + 2x – 8$

d) $x^2 + 3x – 18$

e) $x^2 + 5x – 14$

f) $x^2 + 3x – 40$

g) $x^2 + 5x – 84$

h) $x^2 + 4x – 45$

Resolve into factors.

a) $x^2 – x – 2$

b) $x^2 – x – 6$

c) $x^2 – 2x – 15$

d) $x^2 – 3x – 28$

e) $x^2 – 4x – 12$

f) $x^2 – 3x – 40$

g) $x^2 – 7x – 44$

h) $x^2 – 8x – 65$

Let’s factorise.

a) $x^3 + 15x^2 + 56x$

b) $x^3 + 17x^2 + 42x$

c) $x^4 – 4x^3 – 21x^2$

d) $x^4 – 14x^3 + 45x^2$

e) $x^3 + 13x^2 – 30x$

f) $x^4 + 2x^3 – 8x^2$

g) $x^5 – 18x^4 + 72x^3$

h) $x^5 + 11x^4 – 102x^3$

Let’s factorise.

a) $(x + y)^2 + 5(x + y) + 6$

b) $(a + b)^2 + 14(a + b) + 33$

c) $(p – q)^2 – 11(p – q) + 30$

d) $(x – y)^2 – 9(x – y) – 22$

a) The area of a rectangle is $a^2 + 6a + 8$ sq. units. Find its length and breadth. Also find the perimeter of the rectangle.

b) The area of a rectangular ground is $x^2 + 5x – 36$ sq. unit. If its length and breadth are reduced by 2/2 unit, find the new area of the ground.

Let’s write the following expressions as the product of their factors.

a) $2x(x + 1) + 1(x + 1) =$ ………………..

b) $2x(x – 1) – 1(x – 1) =$ ………………..

c) $3x(2x + 1) + 2(2x + 1) =$ ………………..

d) $3x(2x – 1) – 2(2x – 1) =$ ………………..

e) $2x(3x – 2) + (3x – 2) =$ ………………..

f) $2x(3x + 2) + (3x + 2) =$ ………………..

Let’s find the product and sum or difference of two numbers required for splitting the middle terms.

a) In $2x^2 + 3x + 1$: product of numbers = …….. and sum of numbers = ……..

b) In $3x^2 + 5x + 2$: product of numbers = …….. and sum of numbers = ……..

c) In $4x^2 – 8x + 3$: product of numbers = …….. and sum of numbers = ……..

d) In $5x^2 – 12x + 4$: product of numbers = …….. and sum of numbers = ……..

e) In $2x^2 + 5x – 3$: product of numbers = …….. and difference of numbers = ……..

f) In $3x^2 + 2x – 8$: product of numbers = …….. and difference of numbers = ……..

Let’s resolve into factors.

a) $2x^2 + 5x + 3$

b) $2x^2 + 7x + 5$

c) $3x^2 + 8x + 4$

d) $3x^2 + 4x + 1$

e) $3x^2 + 5x + 2$

f) $3x^2 + 10x + 3$

g) $4x^2 + 12x + 5$

h) $4x^2 + 11x + 7$

i) $2x^2 – 3x + 1$

j) $3x^2 – 10x + 8$

k) $4x^2 – 8x + 4$

l) $8 – 22x + 9x^2$

m) $13 – 17x + 4x^2$

n) $6x^2 – 19x + 14$

o) $3a^2 – 20a + 17$

Let’s factorise.

a) $2x^2 + x – 3$

b) $3x^2 + x – 14$

c) $4x^2 + 5x – 6$

d) $5x^2 + 13x – 6$

e) $6x^2 + 11x – 2$

f) $7x^2 + 25x – 12$

g) $4x^2 + 4x – 3$

h) $6x^2 + 11x – 10$

i) $2a^2 – 5a – 12$

j) $3a^2 – 5a – 8$

k) $4a^2 – 16a – 9$

l) $2a^2 – 7a – 15$

m) $8x^2 – 22x – 21$

n) $6x^2 – 11x – 7$

o) $5x^2 – 13x – 6$

Let’s resolve into factors.

a) $2x^2 + 5xy + 2y^2$

b) $3x^2 + 5xy + 2y^2$

c) $2x^2 – xy – 3y^2$

d) $8x^2 + 2xy – 15y^2$

e) $15x^2 – 17xy – 4y^2$

f) $24a^2 – 38ab + 15b^2$

g) $15a^2 – 28ab + 12b^2$

h) $14a^2 – 38ab + 20b^2$

i) $8a^2 – 45ab – 18b^2$

Let’s factorise:

a) $a^2 + b^2 – c^2 – d^2 + 2ab – 2cd$

b) $p^2 + q^2 – r^2 – s^2 – 2pq – 2rs$

c) $a^2 – b^2 – 2bc – c^2$

d) $x^2 – y^2 + 2yz – z^2$

Let’s resolve into factors:

a) $2(x + y)^2 – 9(x + y) + 10$

b) $3(a + b)^2 – 10(a + b) + 8$

c) $2(p – q)^2 – 3(p – q) – 9$

d) $6(m – n)^2 – 11(m – n) – 7$

a) The area of a rectangular football ground is $(2x^2 + 5x + 3)$ sq. units.

(i) Find the length and breadth of the ground.

(ii) Find the perimeter of the ground.

(iii) If $x = 46text{ m}$, find the actual perimeter of the ground.

b) The area of a rectangular garden is $(4x^2 + 8x + 3)$ sq. ft.

(i) Find the length and breadth of the garden.

(ii) If the length is increased by $5text{ ft.}$ and breadth is decreased by $2text{ ft.}$, find the new area of the garden.