Algebraic Identities Of Polynomials

Algebraic Identities Of Polynomials

You can also read https://www.aplustopper.com/ncert-solutions-for-class-10-maths-chapter-2/ for more solved examples.

People also ask

• Division algorithm for polynomials
• Factorization of polynomials using factor theorem

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