Question

How do you factor \[{y^3} - 125\]?

Answer 1

Hint: In these type polynomials, we will solve and simplify the given expression by using the identity \[{a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)\], 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 3 is known as cubic polynomial. The given polynomial is a cubic polynomial.
Now given equation is \[{y^3} - 125\],
This can be factored by since both terms are perfect cubes, factor using the difference of cubes formula, \[{a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)\]
Now using the identity\[{a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)\], we get,
Now rewrite \[{y^3}\] as \[{\left( y \right)^3}\] and 125 as \[{\left( 5 \right)^3}\], then the equation becomes,
\[ \Rightarrow {\left( y \right)^3} - {5^3}\],
Using the identity we get,
Here \[a = y,b = 5\], then the equation becomes,
\[ \Rightarrow {\left( y \right)^3} - {5^3}\],
Now using the identity \[{a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)\], we get,
\[{y^3} - {5^3} = \left( {y - 5} \right)\left( {{{\left( y \right)}^2} + \left( y \right)\left( 5 \right) + {5^2}} \right)\],
Now simplifying the right hand side we get,
\[{y^3} - 125 = \left( {y - 5} \right)\left( {{y^2} + 5y + 25} \right)\],
The factorising the given polynomial we get \[{y^3} - 125 = \left( {y - 5} \right)\left( {{y^2} + 5y + 25} \right)\],

\[\therefore \]Factorising \[{y^3} - 125\], we get \[\left( {y - 5} \right)\left( {{y^2} + 5y + 25} \right)\] and it is written as \[{y^3} - 125 = \left( {y - 5} \right)\left( {{y^2} + 5y + 25} \right)\].

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,
\[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\],
\[{\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}\],
\[\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}\],
\[{a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)\],
\[{a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)\].