Q1. Add the following:
(i) ab – bc, bc – ca, ca – ab
Ans: (ab – bc) + (bc – ca) + (ca-ab)
= ab – bc + bc – ca + ca – ab
= ab – ab – bc + bc – ca + ca
= 0
(ii) a – b + ab, b – c + bc, c – a + ac
Ans: (a – b + ab) + (b – c + bc) + (c – a + ac)
= a – b + ab + b – c + bc + c – a + ac
= a – a +b – b +c – c + ab + bc + ca
= 0 + 0 + 0 + ab + bc + ca
= ab + bc + ca
(iii) 2p2q2 – 3pq + 4, 5 + 7pq – 3p2q2
Ans: 2p2q2 – 3pq + 4, 5 + 7pq – 3p2q2
= (2p2q2 – 3pq + 4) + (5 + 7pq – 3p2q2)
= 2p2q2 – 3p2q2 – 3pq + 7pq + 4 + 5
= – p2q2 + 4pq + 9
(iv) l2 + m2, m2 + n2, n2 + l2, 2lm + 2mn + 2nl
Ans: (l2 + m2) + (m2 + n2) + (n2 + l2) + (2lm + 2mn + 2nl)
= l2 + l2 + m2 + m2 + n2 + n2 + 2lm + 2mn + 2nl
= 2l2 + 2m2 + 2n2 + 2lm + 2mn + 2nl
Q2. (a) Subtract 4a – 7ab + 3b + 12 from 12a – 9ab + 5b – 3
Ans: (12a – 9ab + 5b – 3) – (4a – 7ab + 3b + 12)
= 12a – 9ab + 5b – 3 – 4a + 7ab – 3b – 12
= 12a – 4a -9ab + 7ab +5b – 3b -3 -12
= 8a – 2ab + 2b – 15
(b) Subtract 3xy + 5yz – 7zx from 5xy – 2yz – 2zx + 10xyz
Ans: (5xy – 2yz – 2zx + 10xyz) – (3xy + 5yz – 7zx)
= 5xy – 2yz – 2zx + 10xyz – 3xy – 5yz + 7zx
=5xy – 3xy – 2yz – 5yz – 2zx + 7zx + 10xyz
= 2xy – 7yz + 5zx + 10xyz
(c) Subtract 4p2q – 3pq + 5pq2 – 8p + 7q – 10 from 18 – 3p – 11q + 5pq – 2pq2 + 5p2q
Ans: (18 – 3p – 11q + 5pq – 2pq2 + 5p2q) – (4p2q – 3pq + 5pq2 – 8p + 7q – 10)
= 18 – 3p – 11q + 5pq – 2pq2 + 5p2q – 4p2q + 3pq – 5pq2 + 8p – 7q + 10
=18+10 -3p+8p -11q – 7q + 5 pq+ 3pq- 2pq2 – 5pq^2 + 5 p^2 q – 4p^2 q
= 28 + 5p – 18q + 8pq – 7pq2 + p2q
Q1: Find the product of the following pairs of monomials.
(i) 4, 7p
Ans: 4 , 7 p = 4 × 7 × p = 28p
(ii) –4p, 7p
Ans: – 4p × 7p = (-4 × 7 ) × (p × p )= -28p2
(iii) –4p, 7pq
Ans: – 4p × 7pq =(-4 × 7 ) (p × pq) = -28p2q
(iv) 4p3, –3p
Ans: 4p3 × – 3p = (4 × -3 ) (p3 × p ) = -12p4
(v) 4p, 0
Ans: 4p × 0 = 0
Q2: Find the areas of rectangles with the following pairs of monomials as their lengths and breadths respectively.
(p, q); (10m, 5n); (20x2, 5y2); (4x, 3x2); (3mn, 4np)
Ans: Area of rectangle = Length x breadth. So, it is the multiplication of two monomials.
The results can be written in square units.
(i) p × q = pq
(ii)10m × 5n = 50mn
(iii) 20x2 × 5y2 = 100x2y2
(iv) 4x × 3x2 = 12x3
(v) 3mn × 4np = 12mn2p
Q3: Complete the table of products.
Ans: The table can be completed as follows.
Q4. Obtain the volume of rectangular boxes with the following length, breadth and height respectively.
Volume of rectangle = length x breadth x height. To evaluate volume of rectangular boxes, multiply all the monomials.
(i) 5a, 3a2, 7a4
Ans: 5a × 3a2 × 7a4 = (5 × 3 × 7) (a × a2 × a4 ) = 105a7
(ii) 2p, 4q, 8r
Ans: 2p × 4q × 8r = (2 × 4 × 8 ) (p × q × r ) = 64pqr
(iii) xy, 2x2y, 2xy2
Ans: y × 2x2y × 2xy2 =(1 × 2 × 2 )( x × x2 × x × y × y × y2 ) = 4x4y4
(iv) a, 2b, 3c
Ans: a x 2b x 3c = (1 × 2 × 3 ) (a × b × c) = 6abc
Q5. Obtain the product of
(i) xy, yz, zx
Ans: xy × yz × zx = x2 y2 z2
(ii) a, – a2, a3
Ans: a × – a2 × a3 = – a6
(iii) 2, 4y, 8y2, 16y3
Ans: 2 × 4y × 8y2 × 16y3 = 1024 y6
(iv) a, 2b, 3c, 6abc
Ans: a × 2b × 3c × 6abc = 36a2 b2 c2
(v) m, – mn, mnp
Ans: m × – mn × mnp = –m3 n2 p
Q1: Identify the terms, their coefficients for each of the following expressions.
(i) 5xyz2 – 3zy
Ans:
Terms | 5xyz2 | –3zy |
Coefficients | 5 | –3 |
(ii) 1 + x + x2
Ans:
Terms | 1 | +x | +x2 |
Coefficients | 1 | 1 | 1 |
(iii) 4x2y2 – 4x2y2z2 + z2
Ans:
Terms | 4x2y2 | –4x2y2z2 | + z2 |
+ z2 | 4 | –4 | 1 |
(iv) 3 – pq + qr – rp
Ans:
Terms | 3 | –pq | + qr | – rp |
Coefficients | 3 | –1 | +1 | –1 |
(v) x/2 +x/2 - xy
Ans:
Terms | x/2 | +y/2 | -xy |
Coefficients | 1/2 | 1/2 | -1 |
(vi) 0.3a – 0.6ab + 0.5b
Ans:
Terms | 0.3a | –0.6ab | 0.5b |
Coefficients | 0.3 | –0.6 | 0.5 |
Q2: Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any of these three categories?
x + y, 1000, x + x2 + x3 + x4, 7 + y + 5x, 2y – 3y2, 2y – 3y2 + 4y3,
5x – 4y + 3xy, 4z – 15z2, ab + bc + cd + da, pqr, p2q + pq2, 2p + 2q
Sol:
Monomials | Binomials | Trinomials |
1000 pqr | x + y 2y – 3y2 4z – 15z2 p2q + pq2 2p + 2q | 7 + y + 5x 2y – 3y2 + 4y3 5x – 4y + 3xy |
Following polynomials do not fit in any of these categories:
x + x2 + x3 + x2 [∵ It has 4 terms.]
ab + bc + cd + da [∵ It has 4 terms.]
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1. What are algebraic expressions? |
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3. What are algebraic identities? |
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