What is the use of algebraic identies

Algebraic Identities Of Polynomials

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Algebraic Identities Of Polynomials Example Problems With Solutions

Example 1:    Expand each of the following

Solution:   (i) We have,

Example 2:    Find the products
(i) (2x + 3y) (2x – 3y)

Solution:    (i) We have,

Example 3:    Evaluate each of the following by using identities
(i) 103 × 97         (ii) 103 × 103
(iii) (97)2             (iv) 185 × 185 – 115 × 115
Solution:    (i) We have,

Example 4:    
Solution:    We have,

Example 5:     
Solution:    We have,

Example 6:    If x + y = 12 and xy = 32, find the value of x2 + y2
Solution:    We have,

Example 7:    Prove that:
2a2 + 2b2 + 2c2 – 2ab – 2bc – 2ca = [(a – b)2 + (b – c)2 + (c – a)2]
Solution:    We have,

Example 8:    If a2 + b2 + c2 – ab – bc – ca = 0, prove that a = b = c.
Solution:    We have,

Example 9:    Write the following in expanded form :
(i) (9x + 2y + z)2            (ii) (3x + 2y – z)2
(iii) (x – 2y – 3z)2         (iv) (–x + 2y + z)2
Solution:    Using the identity

Example 10:    If a2 + b2 + c2 = 20 and a + b + c = 0, find ab + bc + ca.
Solution:    We have,

Example 11:    If a + b + c = 9 and ab + bc + ca = 40, find a2 + b2 + c2.
Solution:    We know that

Example 12:    If a2 + b2 + c2 = 250 and ab + bc + ca = 3, find a + b + c.
Solution:    We know that

Example 13:    Write each of the following in expanded form:
(i) (2x + 3y)3     (ii) (3x ­– 2y)3
Solution:    (i)Replacing a by 2x and b by 3y in the identity

Example 14:    If x + y = 12 and xy = 27, find the value of x3 + y3.
Solution:    We know that

Example 15:    If x – y = 4 and xy = 21, find the value of x3 – y3.
Solution:    We know that

Example 16:    
Solution:    We have,

Example 17:    If a + b = 10 and a2 + b2 = 58, find the value of a3 + b3.
Solution:    We know that

Example 18:    
Solution:    We have,

Example 19:    
Solution:   We know that

Example 20:    If a + b = 10 and ab = 21, find the value of a3 + b3.
Solution:    We know that

Example 21:     If a – b = 4 and ab = 45, find the value of a3 – b3.
Solution:   We have,

Example 22:    If a + b + c = 0, then prove that a3 + b3 + c3 = 3abc
Solution:   We know that

Example 23:    Find the following product:
(x + y + 2z) (x2 + y2 + 4z2 – xy – 2yz – 2zx)
Solution:    We have,

Example 24:     If a + b + c = 6 and ab + bc + ca = 11, find the value of a3 + b3 + c3 – 3abc.
Solution:   We know that

Example 25:    If x + y + z = 1, xy + yz + zx = –1 and xyz = –1, find the value of x3 + y3 + z3.
Solution:    We know that

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