QUESTION

# What is the value of ${\left( {a + b + c} \right)^2}$?

Hint: First define the rules for multiplication of polynomials. Then use these rules to evaluate the given expression by splitting the given polynomial into a product of linear polynomials in three variables.

Complete step-by-step solution -
In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
A monomial is a polynomial which has only one term.
A binomial is a polynomial that is the sum of two terms, each of which is a monomial.
A trinomial is a polynomial consisting of three terms or monomials.
Given in the problem we have the trinomial $\left( {a + b + c} \right)$, where $a,b,c$ are the variables.
We need to find the value of ${\left( {a + b + c} \right)^2}$.
To find the product of two trinomials, we multiply the coefficients together and use the product rule for exponents to multiply any variables with power.
To multiply two polynomials, we need to multiply each term of the first polynomial by each term of the second polynomial and then combine the like terms.
We know that,
${\left( {a + b + c} \right)^2} = \left( {a + b + c} \right)\left( {a + b + c} \right)$
Applying the same rules as stated above to the given trinomial, multiplying each term of first polynomial with the second polynomial, we get
$\Rightarrow {\left( {a + b + c} \right)^2} = a.\left( {a + b + c} \right) + b.\left( {a + b + c} \right) + c.\left( {a + b + c} \right)$
Again, applying the same rules of multiplication of polynomials,
$\Rightarrow {\left( {a + b + c} \right)^2} = a.a + a.b + a.c + b.a + b.b + b.c + c.a + c.b + c.c \\ \Rightarrow {\left( {a + b + c} \right)^2} = {a^2} + ab + ac + ba + {b^2} + bc + ca + cb + {c^2}{\text{ (1)}} \\$
We know that multiplication is commutative in polynomials,
$\Rightarrow ab = ba \\ \Rightarrow ac = ca \\ \Rightarrow bc = cb \\$
Using them in equation (1) and combing he like terms, we get
$\Rightarrow {\left( {a + b + c} \right)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ca$
Hence the value of ${\left( {a + b + c} \right)^2}$ is equal to ${a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ca$.

Note: The above expression derived should be remembered as the formula for square of a trinomial. It is to be noted that polynomials are algebraic expressions having power as rational numbers. Polynomials can be in one variable or more than one variable as in the above case. The above formula can also be derived by taking the sum of two monomials as one term and the remaining one as the other and then applying the algebraic identity of the square of a binomial.