Understanding factorisation is crucial in many areas of math, particularly in algebraic expressions and quadratic equations. By factoring, we break down complex expressions into simpler ones, which makes solving problems and simplifying equations much easier.
So how do you factorise 1 - x^2 ?
To factorise 1 - x^2 , apply the difference of two squares method.
Applying The Difference Of Squares Method
One effective technique for factorising 1 - x^2 is by using the difference of squares method. This involves recognising that the expression can be written as (1)^2 - (x)^2 , which follows the general form of a^2 - b^2 .
We can then apply the formula for this special type of quadratic, which states that it factors into two binomial bracketed expressions: (a + b)(a – b) .
This method works well when dealing with expressions that are in the form of a perfect square minus another perfect square. It simplifies the process and reduces complicated algebraic expressions into simpler forms, making them easier to solve or manipulate.
Therefore, 1 - x^2 can be factorised as follows using the difference of two squares method to:
(1 + x) (1 - x)
Check Answer
How can you check this answer to see if it’s correct?
If you expand the expression using the FOIL method , you will get back the original expression.
The FOIL method represents the expansion of the expression by multiplying the F irst term in each pair of the brackets together:
( \ \fcolorbox{red}{yellow}{1} + x \ )( \ \fcolorbox{red}{yellow}{1} + x \ )
In this case the first term in each bracket is the number 1 , therefore 1 \times 1 = 1 .
The O in FOIL represents the multiplication of the outside terms:
( \ \fcolorbox{red}{yellow}{1} + x \ )( \ 1 \ \fcolorbox{red}{yellow}{$- \ x$} \ )
In this case, the outside terms in each bracket, when multiplied together will produce 1 \times -x = -x .
The next letter I represents I nside or I nner and is the multiplication of the two inner-most terms, as shown here:
( \ 1 \ \fcolorbox{red}{yellow}{$+ \ x$} \ )( \ \fcolorbox{red}{yellow}{$1$} - x \ )
In this case, the inner most terms, when multiplied together will produce x \times 1 = x .
The last latter L represents L ast and is the multiplication of the two last terms in each binomial expression here:
( \ 1 \ \fcolorbox{red}{yellow}{$+ \ x$} \ )( \ 1 \ \fcolorbox{red}{yellow}{$- \ x$} \ )
In this case, the last terms, when multiplied together, will produce x \times -x = -x^2 .
Adding each of the terms from FOIL produces:
1 \ + \ -x \ + \ x \ + \ -x^2
Simplifying this expression by operating on like terms would mean the elimination of -x + x = 0 , therefore, the final result after simplifying would be:
1 - x^2
Which is the original expression you started with. Therefore, you can be confident in your factorisation answer of (1 + x)(1 - x) .
Conclusion
In conclusion, factorisation is an essential skill when working with algebraic expressions and solving quadratic equations. Understanding how to factorise 1 - x^2 can help simplify complex mathematical operations and lead to quicker solutions.
Remember to apply the difference of squares method correctly, avoid common mistakes, and consider the application of factorisation in other areas such as graphing parabolas or simplifying rational expressions.
Then once you have your answer use the FOIL method to check.