1. Identify the terms, their coefficients for each of the following expressions.

(i)

*5xyz*^{2}– 3zyAnswer: Term:

*xyz*^{2}-- Coefficient = 5Term:

*zy*–- Coefficient = 3(ii) 1 +

*x*+*x*^{2}Answer: Term:

*x*– Coefficient = 1 & Term:*x*^{2}– Coefficient = 1(iii)

*4x*y^{2}*y*^{2}– 4x^{2}*z*^{2}^{2}+ z^{2}Answer: 4 is the coefficient for

*x*y^{2}*z*^{2}^{2}1 is the term for z

^{2}(iv)

*3 – pq + qr – rp*Answer: For each term the coefficient is 1

(v) *0.3a – 0.6ab + 0.5b*

Answer: 0.3 is the coefficient for

*a*, for*b*there are two coefficients 0.6 and 0.52. Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any of these three categories?

Answer:

*x + y*: Binomial

1000: Mononomial

*x + x*Polynomial

^{2}+ x^{3}+ x^{4}:*7 + y + 5x:*Binomial

*2y – 3y*

^{2}: Binomial

*2y – 3y*

^{2}+ 4y^{3}: Trinomial

*5x – 4y + 3xy:*Trinomial

*4z – 15z*

^{2}: Binomial

*ab + bc + cd + da:*Polynomial

*pqr:*Mononomial

*p*: Binomial

^{2}q + pq^{2}*2p + 2q:*Binomial

3. Add the following.

*(i) ab – bc, bc – ca, ca – ab*

*a – b + ab, b – c + bc, c – a + ac*

(iii)

*2p*q^{2}*q*^{2}– 3pq + 4, 5 + 7pq – 3p^{2}^{2 }(iv)*l*^{2}+ m^{2}, m^{2}+ n^{2}, n^{2}+ l^{2}, 2lm + 2mn + 2nlAnswer: (ab - bc) + (bc - ca) + (ca-ab)

ab + bc + ca - bc - ca - ab =

= 0

(ii)

*a – b + ab, b – c + bc, c – a + ac*Answer: (a - b + ab) + (b - c + bc) + (c - a + ac)

= a + b + c + ab + bc + ca - b - c - a

= ab + bc + ca

(iii)

*2p*q^{2}*q*^{2}– 3pq + 4, 5 + 7pq – 3p^{2}^{2}= (2p

^{2}q^{2}- 3pq + 4) + (5 + 7pq - 3p^{2}q^{2})= 2p

^{2}q^{2}- 3p^{2}q^{2}- 3pq + 7pq + 4 + 5= - p

^{2}q^{2}+ 4pq + 9(iv)

*l*^{2}+ m^{2}, m^{2}+ n^{2}, n^{2}+ l^{2}, 2lm + 2mn + 2nlAnswer:

*(l*^{2}+ m^{2}) + (m^{2}+ n^{2}) + (n^{2}+ l^{2}) + (2lm + 2mn + 2nl)=

*l*^{2}+ l^{2}+ m^{2}+ m^{2}+ n^{2}+ n^{2}+ 2lm + 2mn + 2nl=

*2l*^{2}+ 2m^{2}+ 2n^{2}+ 2lm + 2mn + 2nl4. (a) Subtract

*4a – 7ab + 3b + 12*from*12a – 9ab + 5b – 3*Answer:

*(12a - 9ab + 5b - 3) - (4a - 7ab + 3b + 12)**= 12a - 9ab + 5b - 3 - 4a + 7ab - 3b - 12*

(signs are reversed after –sign once bracket is opened)

=

*8a - 2ab + 2b - 15*(b) Subtract

*3xy + 5yz – 7zx*from*5xy – 2yz – 2zx + 10xyz*Answer:

*(5xy - 2yz - 2zx + 10xyz) - (3xy + 5yz - 7zx)**= 5xy - 2yz - 2zx + 10xyz - 3xy - 5yz + 7zx*

*= 2xy - 7yz + 5zx + 10xyz*

(c) Subtract

*4p*^{2}q – 3pq + 5pq^{2}– 8p + 7q – 10from

*18 – 3p – 11q + 5pq – 2pq*q^{2}+ 5p^{2}Answer:

*(18 - 3p - 11q + 5pq - 2pq*^{2}+ 5p^{2}q) - (4p^{2}q - 3pq + 5pq^{2}- 8p + 7q - 10)*= 18 - 3p - 11q + 5pq - 2pq*

^{2}+ 5p^{2}q - 4p^{2}q + 3pq - 5pq^{2}+ 8p - 7q + 10*= 28 + 5p - 18q + 8pq - 7pq*q

^{2}- p^{2}5. (a) Add:

*p ( p – q), q ( q – r) and r ( r – p)*Answer:

*(p*^{2}- pq) + (q^{2}- qr) + (r^{2}- pr)*= p*

^{2}+ q^{2}+ r^{2}- pq - qr - pr(b) Add:

*2x (z – x – y)*and*2y (z – y – x)*Answer:

*(2xz - 2x*^{2}- 2xy) + (2yz - 2y^{2}- 2xy)*= 2xz - 4xy + 2yz - 2x*

^{2}- 2y^{2}(c) Subtract:

*3l (l – 4 m + 5 n)*from*4l ( 10 n – 3 m + 2 l )*Answer:

*(40ln - 12lm + 8l*^{2}) - (3l^{2}- 12lm + 15ln)*= 40ln - 12lm + 8l*

^{2}- 3l^{2}- 12lm + 15ln*= 55ln - 24lm + 5l*

^{2}

(d) Subtract:

*3a (a + b + c ) – 2 b (a – b + c)*from*4c ( – a + b + c )**= (-4ac + 4bc + 4c*

^{2}) - (3a^{2}+ 3ab + 3ac)*= -4ac + 4bc + 4c*

^{2}- 3a^{2}- 3ab - 3ac*= -7ac + 4bc + 4c*

^{2}- 3a^{2}- 3ab6. Multiply the binomials.

i) (2x + 5) and (4x – 3)

Answer:

*(2x + 5)(4x - 3)*=

*2x*x*4x*-*2x*x 3 +*5*x*4x*-*5*x*3*=

*8x² - 6x + 20x -15*=

*8x² + 14x -15*(ii) (y – 8) and (3y – 4)

Answer: ( y - 8)(3y - 4)

= y x 3y - 4y - 8 x 3y + 32

= 3y

^{2}- 4y - 24y + 32= 3y

^{2}- 28y + 32(iii)

*(2.5l – 0.5m) and (2.5l + 0.5m)*Answer:

*(2.5l - 0.5m)(2.5l + 0.5)*Using (a+b)(a-b) = a

^{2}- b^{2}We get = 6.25

*l*- 0.25^{2}*m*^{2}(iv)

*(a + 3b) and (x + 5)*Answer: =

*ax + 5a + 3bx + 15b*(v)

*(2pq + 3q*) and^{2}*(3pq – 2q*)^{2}Answer: = 2pq x 3pq - 2pq x 2q

^{2}+ 3q^{2}x 3pq - 3q^{2}x 2q^{2}= 6p

^{2}q^{2}- 4pq^{3}+ 9pq^{3}- 6q^{4}= 6p

^{2}q^{2}- 5pq^{3}- 6p^{4}^{ }

^{}

7. Find the product.

(i) (5 – 2x) (3 + x)

^{}

(ii) (x + 7y) (7x – y)

^{}

iii) (a

^{2}+ b) (a + b^{2})^{}

(iv) (p

^{2}– q^{2}) (2p + q)

^{}

^{}*Answer: = 15 + 5x - 6x - 2x*

^{}

^{2}

= 15 - x - x

^{2}(ii) (x + 7y) (7x – y)

Answer: = 7x

^{2}- xy + 49xy - 7y^{2}= 7x

^{2}- 7y^{2}+ 48xyiii) (a

^{2}+ b) (a + b^{2})Answer: a

^{2}x a + a^{2 }x^{ }b + a x b + b^{3}= a

^{3}+ a^{2}b + ab + b^{3}= a

^{3}+ b^{3}+ a^{2}b + ab(iv) (p

^{2}– q^{2}) (2p + q)Answer: = 2p

^{3}+ p^{2}q - 2pq^{2}- q^{3}= 2p

^{3}- q^{3}+ p^{2}q - 2pq^{2}8. Simplify.

(i) (x

^{2}– 5) (x + 5) + 25Answer: = x

^{3}+ 5x^{2}- 5x - 25 + 25= x3 + 5x

^{2}-5x(ii) (a

^{2}+ 5) (b^{3}+ 3) + 5Answer: a

^{2}b^{3}+ 3a^{2}+ 5b^{3}+ 15 + 5= a

^{2}b^{3}+ 5b^{3}+ 3a^{2}+ 20(iv) (a + b) (c – d) + (a – b) (c + d) + 2 (ac + bd)

Answer: = (ac - ad + bc - bd) + (ac + ad - bc - bd) + (2ac + 2bd)

= ac - ad + bc - bd + ac + ad - bc - bd + 2ac + 2bd

= 4ac

(v) (x + y)(2x + y) + (x + 2y)(x – y)

Answer: 2x

^{2}+ xy + 2xy + y^{2}+ x^{2}- xy + 2xy - 2y^{2}= 3x

^{2}+ 4xy - y^{2}^{ }

(vi) (x + y)(x

^{2}– xy + y^{2})Answer: = x

^{3}- x^{2}y + xy^{2}+ x^{2}y - xy^{2}+ y^{3}= x

^{3}+ y^{3}^{ }

(vii) (1.5x – 4y)(1.5x + 4y + 3) – 4.5x + 12y

Answer: = 2.25x

^{2}+ 6xy + 4.5x - 6xy - 16y^{2}- 12y - 4.5x + 12y= 2.25x

^{2}- 16y^{2}^{ }

(viii) (a + b + c)(a + b – c)

Answer: = a

^{2}+ ab - ac + ab + b^{2}- bc + ac + bc - c^{2}= a

^{2}+ b^{2}- c^{2}+ 2ab9 Use a suitable identity to get each of the following products.

(i) (x + 3) (x + 3)

Answer: Using (a + b)

^{2}= a^{2 }+ 2ab + b^{2}we get the following equation:= x

^{2}+ 6x + 9(ii) (2y + 5) (2y + 5)

Answer: 4y

^{2}+ 20y + 25(iii) (2a – 7) (2a – 7)

Answer: Using (a - b)

^{2}= a^{2 -}2ab + b^{2}we get the following equation:= 4a

^{2}- 28a + 4910. Use the identity (x + a) (x + b) = x

^{2 }+ (a + b) x + ab to find the following products.(i) (x + 3) (x + 7)

Answer: x

^{2}+ (3+7)x + 21= x

^{2}+ 10x + 21(ii) (4x + 5) (4x + 1)

= 16x

^{2}+ (5 + 1)4x + 5= 16x

^{2}+ 24x + 5(iii) (4x – 5) (4x – 1)

= 16x

^{2}+ (-5-1)4x + 5= 16x

^{2}- 20x + 5(iv) (4x + 5) (4x – 1)

= 16x

^{2}+ (5-1)4x - 5= 16x

^{2}+16x - 5(v) (2x + 5y) (2x + 3y)

= 4x

^{2}+ (5y + 3y)4x + 15y^{2}= 4x

^{2}+ 32xy + 15y^{2}^{ }

(vi) (2a

^{2}+ 9) (2a^{2}+ 5)= 4a

^{4}+ (9+5)2a^{2}+ 45= 4a

^{4}+ 28a^{2}+ 45(vii) (xyz – 4) (xyz – 2)

= x

^{2}y^{2}z^{2}+ (-4 -2)xyz - 8= x

^{2}y^{2}z^{2}- 6xyz - 8======================================================================

Variable and expression are must be understood while solving it's related problems.A symbol for a number which is usually a letter like x or y called as Variable and a number on its own is called a Constant.When these two or more variables are joined together with the help of the symbols are called expressions.

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