 The Expansion of (a ± b ± c)^2 can be read as the whole squares of a plus or minus b plus or minus c. It is the formula that is used to find the square of sum or difference among three constant terms such as a, b, c. To derive the Expansion of Powers of Binomials and Trinomials we need to use some set of formulas. Let us learn the derivations of (a + b + c)² and (a – b – c)² with some suitable examples from this page.

Also, Read: Expansion of (a ± b)^2

Expansion of (a ± b ± c)^2 | Expansion of Trinomial with Power 2

Expansion with three constant terms like a, b, c is known as the trinomial expressions. Check the below section to know how to derive (a + b + c)² and (a – b – c)² and where it is used.

Derivations of (a + b + c)²:
The derivation of (a + b + c)² is explained in detail here.
(a + b + c)² = (a + b + c) (a + b + c)
(a + b + c)² = a (a + b + c) + b (a + b + c) + c (a + b + c)
(a + b + c)² = a² + ab + ac + ab + b² + bc + ac + bc + c²
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
Thus the formula of (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Derivation of (a – b – c)²:
The derivation of the trinomial expression (a – b – c)² is explained in detail here.
(a – b – c)² = (a – b – c) (a – b – c)
(a – b – c)² = a (a – b – c) – b (a – b – c) -c (a – b – c)
(a – b – c)² = a² – ab – ac – ab + b² + bc – ac – bc + c²
(a – b – c)² = a² + b² + c² – 2ab + 2bc – 2ac
(a – b – c)² = a² + b² + c² – 2(ab – bc + ca)
Thus the formula of (a – b – c)² = a² + b² + c² – 2(ab – bc + ca)

Expansion of (a ± b ± c)^2 Examples

Let us see some suitable examples on the expansion of (a ± b ± c)² to know how and where to use the formulas of (a + b + c)² and (a – b – c)² in algebra.

Example 1.
Find the value of (a + b + c)² if the values of a, b, c are 3, 4, 2.
Solution:
Given the values of a, b, c are 3, 4, 2.
We have to substitute the values of a, b, c in the formula (a + b + c)²
We know that
The formula of (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
(3 + 4 + 2)² = 3² + 4² + 2² + 2(3(4) + (4)2 + 2(3))
= 9 + 16 + 4 + 2(12 + 8 + 6)
= 29 + 2(26)
= 29 + 52
= 81

Example 2.
Expand (2a + 4b + 3c)² by using the (a + b + c)² formula.
Solution:
Given the trinomial expression (2a + 4b + 3c)²
We know that
The formula of (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
We have to substitute the values of a, b, c in the formula (a + b + c)²
(2a + 4b + 3c)² = (2a)² + (4b)² + (3c)² + 2((2a)(4b) + (4b)3c + 3c(2a))
(2a + 4b + 3c)² = 4a² + 16b² + 9c² + 2(8ab + 12bc + 6ac)
(2a + 4b + 3c)² = 4a² + 16b² + 9c² + 16ab + 24bc + 12ac
Thus the expansion of (2a + 4b + 3c)² is 4a² + 16b² + 9c² + 16ab + 24bc + 12ac

Example 3.
Find the value of (a – b – c)² if the values of a, b, c are 5, 3, 2.
Solution:
Given the values of a, b, c are 5, 3, 2.
We know that
The formula of (a – b – c)² = a² + b² + c² – 2(ab – bc + ca)
We have to substitute the values of a, b, c in the formula (a – b – c)²
(5 – 3 – 2)² = 5² + 3² + 2² – 2((5)(3) – (3) (2) + (2)(5))
(5 – 3 – 2)² = 25 + 9 + 4 – 2(15 – 6 + 10)
(5 – 3 – 2)² = 38 – 2(19)
(5 – 3 – 2)² = 38 – 38 = 0
Thus the value of (a – b – c)² if the values of a, b, c are 5, 3, 2 is 0.

Example 4.
Expand (3x + 2y + 5z)² by using the (a + b + c)² formula.
Solution:
Given the trinomial expression (3x + 2y + 5z)²
We know that
The formula of (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
We have to substitute the values of a, b, c in the formula (a + b + c)²
(3x + 2y + 5z)² = (3x)² + (2y)² + (5z)² + 2((3x)(2y) + (2y)(5z) + (5z)(3x))
(3x + 2y + 5z)² = 9x² + 4y² + 25z² + 2(6xy + 10yz + 15zx)
(3x + 2y + 5z)² = 9x² + 4y² + 25z² + 12xy + 20yz + 30zx
Thus the expansion of (3x + 2y + 5z)² is 9x² + 4y² + 25z² + 12xy + 20yz + 30zx

Example 5.
Expand (8x – 4y – 6z)² by using the (a – b – c)² formula.
Solution:
Given the trinomial expression (8x – 4y – 6z)²
We know that
The formula of (a – b – c)² = a² + b² + c² – 2(ab – bc + ca)
We have to substitute the values of a, b, c in the formula (a – b – c)²
(8x – 4y – 6z)² = (8x)² + (4y)² + (6z)² – 2((8x)(4y) – (4y)(6z) + (6z)(8x))
(8x – 4y – 6z)² = 64x² + 16y² + 36z² – 2(32xy) – 24yz + 48zx)
(8x – 4y – 6z)² = 64x² + 16y² + 36z² – 64xy – 48yz + 96zx
Thus the expansion of (8x – 4y – 6z)² is 64x² + 16y² + 36z² – 64xy – 48yz + 96zx

FAQs on Expansion of (a ± b ± c)²

1. What is the Expansion of the (a – b – c)² Formula?

The Expansion of (a – b – c)² formula is a² + b² + c² – 2(ab – bc + ca) or a² + b² + c² – 2ab + 2bc – 2ac

2. What is the Expansion of (a + b + c)² Formula?

The Expansion of (a + b + c)² formula is a² + b² + c² + 2(ab + bc + ca) or a² + b² + c² + 2ab + 2bc + 2ac

3. What type of expression is Expansion of (a ± b ± c)^2?

The Expansion of (a ± b ± c)^2 is a trinomial expression because it consists of three constant terms.