## 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 x^{2} + y^{2}

**Solution: **We have,

**Example 7: **Prove that:

2a^{2} + 2b^{2} + 2c^{2} – 2ab – 2bc – 2ca = [(a – b)^{2} + (b – c)^{2} + (c – a)^{2}]

**Solution: **We have,

**Example 8: **If a^{2} + b^{2} + c^{2} – 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 a^{2} + b^{2} + c^{2} = 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 a^{2} + b^{2} + c^{2}.

**Solution: **We know that

**Example 12: **If a^{2} + b^{2} + c^{2} = 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 x^{3} + y^{3}.

**Solution: **We know that

**Example 15: **If x – y = 4 and xy = 21, find the value of x^{3} – y^{3}.

**Solution: **We know that

**Example 16: **

**Solution: **We have,

**Example 17: **If a + b = 10 and a2 + b2 = 58, find the value of a^{3} + b^{3}.

**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 a^{3} + b^{3}.

**Solution: **We know that

**Example 21: **If a – b = 4 and ab = 45, find the value of a^{3} – b^{3}.

**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) (x^{2} + y^{2} + 4z^{2} – xy – 2yz – 2zx)

**Solution: **We have,

**Example 24: ** If a + b + c = 6 and ab + bc + ca = 11, find the value of a^{3} + b^{3} + c^{3} – 3abc.

**Solution: **We know that

**Example 25: **If x + y + z = 1, xy + yz + zx = –1 and xyz = –1, find the value of x^{3} + y^{3} + z^{3}.

**Solution: **We know that