Algebraic Expressions and Identities

Using (x + a) (x + b) = x² + (a + b) x + ab, find

(i) 103 x 104 (ii) 5.1 x 5.2 (iii) 103 x 98 (iv) 9.7 x 9.8

Using a² - b² = (a + b) (a - b), find

(i) 51² - 49² (ii) (1.02)² - (0.98)² (iii) 153² - 147²

(iv) 12.1² - 7.9²

Using identities, evaluate.

(i) 71² (ii) 99² (iii) 102² (iv) 998²

(v) (5.2)² (vi) 297 x 303 (vii) 78 x 82

(viii) 8.9² (ix) 1.05 x 9.5

(i) (3x + 7)² - 84x = (3x - 7)² (ii) (9p - 5q)² + 180pq = (9p + 5q)²

(iii) 

(iv) (4pq + 3q)² - (4pq - 3q)² = 48pq²

(v) (a - b) (a + b) + (b - c) (b + c) + (c - a) (c + a) = 0

Simplify.

(i) (a² - b²)² (ii) (2x +5)² - (2x - 5)²

(iii) (7m - 8n)² + (7m + 8n)² (iv) (4m + 5n)² + (5m + 4n)²

(v) (2.5p - 1.5q)² - (1.5p - 2.5q)²

(vi) (ab + bc)² - 2ab²c (vii) (m² - n²m)² + 2m³n²

Find the following squares by suing the identities.

(i) (b - 7)² (ii) (xy + 3z)² (iii) (6x² - 5y)²

(iv) 

  (v) (0.4p - 0.5q)² (vi) (2xy + 5y)²

Use the identity (x + a) (x + b) = x² + (a + b)x + ab to find the following products.

(i) (x + 3) (x + 7) (ii) (4x +5) (4x + 1)

(iii) (4x - 5) (4x - 1) (iv) (4x + 5) (4x - 1)

(v) (2x +5y) (2x + 3y) (vi) (2a² +9) (2a² + 5)

(vii) (xyz - 4) (xyz - 2)

(i) (x + 3) (x + 3) (ii) (2y + 5) (2y + 5)

(iii) (2a - 7) (2a - 7) (iv) 

(v) (1.1m - 0.4) (1.1 m + 0.4) (vi) (a² + b²) ( - a² + b²)

(vii) (6x - 7) (6x + 7) (viii) ( - a + c) ( - a + c)

(ix)

  (x) (7a - 9b) (7a - 9b)

Simplify.

(i) (x² - 5) (x + 5) + 25

(ii) (a² + 5) (b³ + 3) + 5

(iii) (t + s²) (t² - s)

(iv) (a + b) (c - d) + (a - b) (c + d) + 2 (ac + bd)

(v) (x + y) (2x + y) + (x + 2y) (x - y)

(vi) (x + y) (x² - xy + y²)

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

(viii) (a + b + c) (a + b - c)

Find the product.

(i) (5 - 2x) (3 + x) (ii) (x + 7y) (7x - y)

(iii) (a² + b) (a + b²) (iv) (p² - q²) (2p + q)

(i) (2x + 5) and (4x - 3) (ii) (y - 8) and (3y - 4)

(iii) (2.5l - 0.5m) and (2.5l + 0.5m) (iv) (a + 3b) and (x + 5)

(v) (2pq + 3q²) and (3pq - 2q²)

(vi) 

(a) Add: p (p - q), q (q _­­­- r) and r (r ­- p)

(b) Add: 2x (z - x - y) and 2y (z - y - x)

(c) Subtract: 3l (l - 4m + 5n) from 4l (10n - 3m + 2l)

(d) Subtract: 3a (a + b + c) - 2b (a - b + c) from 4c (- a + b + c)

a) Simplify 3x (4x - 5) + 3 and find its values for (i) x = 3, (ii) .

(b) a (a² + a + 1) + 5 and find its values for (i) a = 0, (ii) a = 1, (iii) a = - 1.

(i) (a²) × (2a²²) × (4a²⁶)

(ii) 

(iii) 

(iv) x × x² × x³ × x⁴

Complete the table

Carry out the multiplication of the expressions in each of the following pairs.

(i) 4p, q + r (ii) ab, a - b (iii) a + b, 7a²b²

(iv) a² - 9, 4a (v) pq + qr + rp, 0

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

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

Complete the table of products.

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)

Find the product of the following pairs of monomials.

(i) 4, 7p (ii) - 4p, 7p (iii) - 4p, 7pq

(iv) 4p³, - 3p (v) 4p, 0

(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

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

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

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

(i) 5xyz² - 3zy

(ii) 1 + x + x²

(iii) 4x²y² - 4x²y²z² + z²

(iv) 3 - pq + qr - rp

(v) 

(vi) 0.3a - 0.6ab + 0.5b