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:
$$ (1 + x)(1 - x) $$
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 First, Outer, Inner, and Last terms.
First
$$ 1 \times 1 = 1 $$
Outer
$$ 1 \times (-x) = -x $$
Inner
$$ x \times 1 = x $$
Last
$$ x \times (-x) = -x^2 $$
Adding all the FOIL terms together:
$$ 1 + (-x) + x + (-x^2) $$
Simplifying like terms:
$$ 1 - x + x - x^2 = 1 - x^2 $$
Which is the original expression. Therefore, the factorisation
$$ (1 + x)(1 - x) $$
is correct.
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 it.