Question

How do you multiply \[{\left( {x + 4} \right)^3}\]?

Answer 1

Hint: As the given expression is of the form \[\left( {x + a} \right)\left( {x + b} \right)\left( {x + c} \right)\], hence to multiply the given expression we need to expand the terms as \[{x^3} + \left( {a + b + c} \right){x^2} + \left( {ab + ac + bc} \right)x + abc\], and according to this we have the values of a, b and c as: \[a = b = c = 4\], hence substitute these values in the expansion and simplify the terms.

Formula used:
\[\left( {x + a} \right)\left( {x + b} \right)\left( {x + c} \right) = {x^3} + \left( {a + b + c} \right){x^2} + \left( {ab + ac + bc} \right)x + abc\]

Complete step by step answer:
Let us write the given expression:
\[{\left( {x + 4} \right)^3}\] ………………….. 1
The given expression is factors of the form:
\[\left( {x + a} \right)\left( {x + b} \right)\left( {x + c} \right)\]
This is expanded as:
\[ \Rightarrow {x^3} + \left( {a + b + c} \right){x^2} + \left( {ab + ac + bc} \right)x + abc\] …………………… 2
Hence, we have equation 1 as:
\[{\left( {x + 4} \right)^3} = \left( {x + 4} \right)\left( {x + 4} \right)\left( {x + 4} \right)\]
Here,
\[a = b = c = 4\]
Now, let us expand the terms according to the equation 2 as:
\[{x^3} + \left( {a + b + c} \right){x^2} + \left( {ab + ac + bc} \right)x + abc\]
Substitute the value of a, b and c we get:
\[ = {x^3} + \left( {4 + 4 + 4} \right){x^2} + \left( {\left( 4 \right)\left( 4 \right) + \left( 4 \right)\left( 4 \right) + \left( 4 \right)\left( 4 \right)} \right)x + \left( 4 \right)\left( 4 \right)\left( 4 \right)\]
Simplifying the terms, we have:
\[ = {x^3} + 12{x^2} + \left( {16 + 16 + 16} \right)x + {4^3}\]
\[ = {x^3} + 12{x^2} + 48x + 64\]

Therefore,
\[{\left( {x + 4} \right)^3} = {x^3} + 12{x^2} + 48x + 64\]

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.
It is known that there are different types of polynomial based on its degree like linear, binomial, quadratic, trinomial, etc. The steps to multiply polynomials are the same for all the types. There are two types of multiplication of polynomials i.e., Multiplication of Binomial by a Binomial and Multiplication of a Binomial by a Trinomial.

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 \[{\left( {x + y} \right)^n}\].