Question

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

Answer 1

Hint: In these type polynomials, we will solve the given expression by adding and subtracting \[{x^2}\] and will simplify the given expression by using the identities \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\] and \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\], and we will get the required simplified term of the given expression.

Complete step-by-step solution:
The given expression is a polynomials which are algebraic expressions that are composed of two or more algebraic terms, the algebraic terms are constant, variables and exponents, there are different types of polynomials which are differentiated with the help of degree of the polynomial.
The polynomial of degree 4 is known as quadratic polynomial. The given polynomial is a biquadratic polynomial.
Now given equation is \[{x^4} + 3{x^2} + 4\],
Now add and subtract \[{x^2}\] from the equation, we get,
\[{x^4} + 3{x^2} + 4 + {x^2} - {x^2}\],
Now adding the like terms we get,
\[ \Rightarrow {x^4} + 4{x^2} + 4 - {x^2}\],
\[ \Rightarrow \left( {{x^4} + 4{x^2} + 4} \right) - {x^2}\],
Now the expression \[{x^4} + 4{x^2} + 4\] can be written as a perfect square we get,
Now using the identity \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\],
Here \[a = {x^2},b = 2\], the expression becomes,
\[ \Rightarrow \]\[{x^4} + 4{x^2} + 4 = {\left( {{x^2} + 2} \right)^2}\],
Now the given expression becomes,
\[ \Rightarrow {\left( {{x^2} + 2} \right)^2} - {x^2}\],
Now this in form of identity \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\], so using the identity,
Here \[a = {x^2} + 2,b = x\],
Now substituting the values in the identity we get,
\[ \Rightarrow {\left( {{x^2} + 2} \right)^2} - {x^2} = \left( {{x^2} + 2 - x} \right)\left( {{x^2} + 2 + x} \right)\],
The factorising the given polynomial we get \[{x^4} + 3{x^2} + 4 = \left( {{x^2} - x + 2} \right)\left( {{x^2} + x + 2} \right)\],

\[\therefore \]Factorising \[{x^4} + 3{x^2} + 4\], we get \[\left( {{x^2} - x + 2} \right)\left( {{x^2} + x + 2} \right)\] and it is written as \[{x^4} + 3{x^2} + 4 = \left( {{x^2} - x + 2} \right)\left( {{x^2} + x + 2} \right)\].

Note: Factorization is a process which is necessary to simplify the algebraic expressions and is used to solve the higher degree equations. It is the inverse procedure of the multiplication of the polynomials. The algebraic expression is said to be in a factored form only when the whole expression is an indicated product. Factoring polynomials using the identities is done by using the algebraic identities, when it comes to factorization, the commonly used identities are as follows,
\[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\],
\[{\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}\],
\[\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}\],
\[{a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)\],
\[{a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)\].