Question

How do you find the product of \[{\left( {x + 8} \right)^2}\]?

Answer 1

Hint: As the given equation is of the form \[{\left( {a + b} \right)^2}\], hence to find the product of the given equation we need to expand the terms as \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\], and according to this we have the values of a, b as: \[a = x\] and \[b = 8\], hence substitute these values in the expansion and simplify the terms to get the product.

Formula used:
\[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]

Complete step by step solution:
Let us write the given equation:
\[{\left( {x + 8} \right)^2}\]
The given expression is of the form \[{\left( {a + b} \right)^2}\] and this is expanded as \[{a^2} + 2ab + {b^2}\].
Hence, use the formula for the square of a binomial as:
\[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
So, the square of a sum consists of the square of the first term, plus the square of the second, plus twice the product of the two terms.
Hence, as per the given equation let us apply the formula:
\[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
As, per given equation, we have the values of a, b as: \[a = x\] and \[b = 8\],hence we have:
\[ \Rightarrow {\left( {x + 8} \right)^2} = {x^2} + 2\left( x \right)\left( 8 \right) + {\left( 8 \right)^2}\]
Simplifying the terms, we get:
\[ \Rightarrow {\left( {x + 8} \right)^2} = {x^2} + 16x + 64\]

Additional information:
To multiply polynomials, first, multiply each term in one polynomial by each term in the other polynomial using distributive law. Then, simplify the resulting polynomial by adding or subtracting the like terms. It should be noted that the resulting degree after multiplying two polynomials will be always more than the degree of the individual polynomials.

Note: We must know that when we multiply two terms together, we must multiply the coefficient (numbers) and add the exponents. When multiplying two binomials, the distributive property is used because in this method multiplication is carried out by multiplying each term of the first factor to the second factor. So, multiplication of two two-term polynomials is expressed as a trinomial. A binomial can be raised to the \[{n^{th}}\] power and expressed in the form \[{\left( {x + y} \right)^n}\].