Factorise
a.) [tex]x^3-3 x^2-x+3[/tex]
Answer (1)
Group terms: ( x 3 − 3 x 2 ) + ( − x + 3 ) .
Factor out common factors: x 2 ( x − 3 ) − 1 ( x − 3 ) .
Factor out the common binomial: ( x − 3 ) ( x 2 − 1 ) .
Factorise the difference of squares: ( x − 3 ) ( x − 1 ) ( x + 1 ) .
The final factorised form is ( x − 3 ) ( x − 1 ) ( x + 1 ) .
Explanation
Understanding the Problem We are asked to factorise the polynomial x 3 − 3 x 2 − x + 3 . Factoring involves expressing the polynomial as a product of simpler polynomials or factors. We will use factoring by grouping to achieve this.
Grouping Terms We group the terms in pairs: ( x 3 − 3 x 2 ) + ( − x + 3 ) .
Factoring out Common Factors We factor out the common factor from each pair. From the first pair, we can factor out x 2 , and from the second pair, we can factor out − 1 . This gives us: x 2 ( x − 3 ) − 1 ( x − 3 ) .
Factoring out the Common Binomial Now we observe that ( x − 3 ) is a common binomial factor. We factor it out: ( x − 3 ) ( x 2 − 1 ) .
Factoring the Difference of Squares We recognise that x 2 − 1 is a difference of squares, which can be further factored as ( x − 1 ) ( x + 1 ) . Therefore, the fully factorised form is ( x − 3 ) ( x − 1 ) ( x + 1 ) .
Final Answer The factorised form of the given polynomial is ( x − 3 ) ( x − 1 ) ( x + 1 ) .
Examples
Factoring polynomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or circuits. Imagine you are designing a rectangular garden bed and you know the area can be represented by the expression x 3 − 3 x 2 − x + 3 . By factoring this expression into ( x − 3 ) ( x − 1 ) ( x + 1 ) , you can determine possible dimensions for the garden bed in terms of x . This helps in planning and optimising the garden layout.
