Question

How do you factor \[{x^4} + 2{x^3} - 8x - 16\] ?

Answer 1

Hint: We are asked to factorize the given expression. First observe each term and check whether there is any relation between the terms. Take common if possible. Here you will also need to use the formula of \[\left( {{a^3} - {b^3}} \right)\].

Complete step by step solution:
Given, the expression \[{x^4} + 2{x^3} - 8x - 16\].
Let \[A = {x^4} + 2{x^3} - 8x - 16\] ………………(i)
We are asked to factorize this expression. First we observe that \[16 = 8 \times 2\] so there is a connection between the first two terms and the last two terms. Therefore, taking \[{x^3}\] common from first two terms and \[8\] common from last two terms we get,
\[A = {x^3}\left( {x + 2} \right) - 8\left( {x + 2} \right)\]
\[ \Rightarrow A = \left( {x + 2} \right)\left( {{x^3} - 8} \right)\] ……………...(ii)

We can \[8\] as \[{2^3}\] so, writing \[{2^3}\] in place of \[8\] we get,
\[A = \left( {x + 2} \right)\left( {{x^3} - {2^3}} \right)\]
Now, in the second term in \[A\] is in the form \[\left( {{a^3} - {b^3}} \right)\].
The formula for \[\left( {{a^3} - {b^3}} \right)\] is,
\[\left( {{a^3} - {b^3}} \right) = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)\] ………………(iii)

Here \[a = x\] and \[b = 2\]. Using the formula from equation (iii) we get,
\[\left( {{x^3} - {2^3}} \right) = \left( {x - 2} \right)\left( {{x^2} + 2x + {2^2}} \right)\]
\[ \Rightarrow \left( {{x^3} - {2^3}} \right) = \left( {x - 2} \right)\left( {{x^2} + 2x + 4} \right)\]
Putting this value of \[\left( {{x^3} - {2^3}} \right)\] in equation (ii) we get,
\[A = \left( {x + 2} \right)\left( {x - 2} \right)\left( {{x^2} + 2x + 4} \right)\]

Therefore, factoring \[{x^4} + 2{x^3} - 8x - 16\] we get, \[\left( {x + 2} \right)\left( {x - 2} \right)\left( {{x^2} + 2x + 4} \right)\].

Note: The given expression is a polynomial of degree four. By degree we mean the highest power of the variable as here the highest power of \[x\] is four. On the basis of degree polynomials can be classified as constant polynomial, linear polynomial, quadratic polynomial, cubic polynomial and quartic polynomial. Constant polynomial is a polynomial of zero degree. Linear polynomial is a polynomial of degree one, quadratic polynomial is a polynomial of degree two, cubic polynomial is a polynomial of degree three and quartic polynomial is a polynomial of degree four. The given polynomial is a polynomial of degree four, so it is a quartic polynomial.