(m+3)(m-3)(m^2+9)
Using suitable identities evaluate
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Answer:
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2. Classify the following polynomials as monomials, bionomials, trinomlals. 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. The given polynomials are classified as under:
Monomials: 1000, pqr
Binomials: x + y, 2y – 3y2, 4z – 15z2, p2q + pq2, 2p + 2q.
Trinomials: 7 + y + 5x, 2y – 3y2 + 4y3,
5x – 4y + 3xy.
Polynomials that do not fit in any categories:
x + x2 + x3 + x4, ab + bc + cd + da
3. Add the following:
(i) ab – bc, bc – ca, ca – ab
(ii) a – b + ab, b – c + bc, c – a + ac
(iii) 2p2q2 – 3pq + 4, 5 + 7pq – 3p2q2
(iv) l2 + m2, m2 + n2, n2 + l2, 2lm + 2mn + 2nl
Sol. (i) Writing the given expressions in separate rows with like terms one below the other, we have
(ii) Writing the given expressions in separate rows with like terms one below the other, we have
(iii) Writing the given expressions in separate rows with like terms one below the other, we have
(iv) Writing the given expressions in separate rows with like terms one below the other, we have
4. (a) Subtract 4a – 7ab + 3b + 12
from 12a – 9ab + 5b – 3
(b) Subtract 3xy + 5yz – 7zx
from 5xy – 2yz – 2zx + 10xyz
(c) Subtract 4p2q – 3pq + 5pq2 – 8p + 7q – 10
from 18 – 3p – 11q + 5pq – 2pq2 + 5p2q.
Sol. Rearranging the terms of the given expressions, changing the sign of each term of the expression to be subtracted and adding the two expressions, we get.
(b) Rearranging the terms of the given expressions, changing the sign of each term of the expression to be subtracted and adding the two expressions, we get.
(c) Rearranging the terms of the given expressions, changing the sign of each term of the expression to be subtacted and adding the two expressions, we get.
EXERCISE : 9.2
1. Find the product of the following pairs of monomials
(i) 4, 7p (ii) – 4p, 7p
(iii) – 4p, 7pq (iv) 4p3, – 3p
(v) 4p, 0
Sol. (i) 4 × 7p = (4 × 7) × p = 28p
(ii) – 4p × 7p = (– 4 × 7) × (p × p)
= 28 p1+1
= 28p2q
(iii) –4p × 7pq = (–4 × 7) × (p × p × q)
= – 28p1+1q
= – 28p2q
(iv) 4p3 × (–3p) = {4 × (–3)} × (p3 × p)
= – 12 × p3+1
= –12p4
(v) 4p × 0 = (4 × 0) × (p)
= 0 × p
= 0.
2. 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)
Sol. We know that the area of a rectangle = l × b, where l = length and b = breadth.
Therefore, the areas of rectangles with pair of monomials (p, q); (10m, 5n); (20x2, 5y2);
(4x, 3x2) and (3mn, 4np) as their lengths and breadths are given by
p × q = pq
10m × 5n = (10 × 5) × (m × n) = 50mn
20x2 × 5y2 = (20 × 5) × (x2 × y2)
= 100x2y2
4x × 3x2 = (4 × 3) × (x × x2)
= 12x3
and, 3mn × 4np = (3 × 4) × (m × n × n × p)
= 12mn2p
3. Complete the table of products:
Sol. Complete table is as under:
4. Obtain the volume of rectangular boxes with the following length, breadth and height respectively:
(i) 5a, 3a2, 7a4 (ii) 2p, 4q, 8r
(iii) xy, 2x2y, 2xy2 (iv) a, 2b, 3c
Sol. (i) Required volume = 5a × 3a2 × 7a4
= (5 × 3 × 7) × (a × a2 × a4)
= 105a1+2+4 = 105a7
(ii) Required volume = 2p × 4q × 8r
= (2 × 4 × 8) × (p × q × r)
= 64 pqr.
(iii) Required volume = xy × 2x2y × 2xy2
= (1 × 2 × 2) × (xy × x2y × xy2)
= (4) × (x1+2+1 × y1+1+2)
= 4x4y4
(iv) Required volume = a × 2b × 3c
= (1 × 2 × 3) × (a × b × c)
= 6abc
5. Obtain the product of
(i) xy, yz, zx (ii) a, –a2, a3
(iii) 2, 4y, 8y2, 16y3 (iv) a, 2b, 3c, 6abc
(v) m, – mn, mnp
Sol. (i) xy × yz × zx = x × x × y × y × z × z
= x1+1 xy1+1 × y1+1 × z1+1
= x2y2z2
(ii) a × (–a2) × a3
= [1 × (–1) × 1] × (a × a × a × a × a × a)
= (–1) × (a6)
= –a6
(iii) 2 × (4y) × 8y2 × 16y3
= (2 × 4 × 8 × 16) × (y × y2 × y3)
= (1024) × (y1 + 2 + 3)
= 1024y6
(iv) a × 2b × 3c × 6abc
= (2 × 3 × 6) × (a × b × c × abc)
= (36) × (a1+1 × b1+1 × c1+1)
= 36a2b2c2
(v) m × – mn × mnp
= (1 × – 1 × 1)× (m × m × m × n × n × p)
= –1 × m3 × n2 × p
= – m3n2p
EXERCISE : 9.3
1. Carry out the multiplication of the expression in each of the following pairs:
(i) 4p, q + r
(ii) ab, a – b
(iii) a + b, 7a2b2
(iv) a2 – 9, 4a
Sol. (i) 4p × (q + r)
= 4p × q + 4p × r
= 4pq + 4pr
(ii) ab × (a – b)
ab × a – ab × (b)
= a2b – ab2
(iii) (a + b) × (7a2b2)
= 7a2b2 × a + 7a2b2 × b
= 7a3b2 + 7a2b3
(iv) (a2
– 9) × 4a
= a2
× 4a – 9 × 4a
4a3 – 36a
(v) (pq + qr + rp) × 0 = 0