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Questions and Answers
What is the purpose of factorisation in algebra?
What is the purpose of factorisation in algebra?
- To simplify expressions and solve equations (correct)
- To increase the number of terms in an expression
- To confuse students
- To complicate expressions
In factorisation by grouping, why do we rearrange terms in an expression?
In factorisation by grouping, why do we rearrange terms in an expression?
- To make the expression longer
- To factor out a common term from each group (correct)
- To keep the expression the same
- To confuse the reader
What is the factored form of the difference of squares expression "(x^2 - 16)"?
What is the factored form of the difference of squares expression "(x^2 - 16)"?
- (x + 4)(x - 4) (correct)
- (x - 4)(x + 4)
- (x + 4)(x + 4)
- (x - 4)(x - 4)
What is the formula for factoring the sum of cubes expression "a^3 + b^3"?
What is the formula for factoring the sum of cubes expression "a^3 + b^3"?
What is the result of factoring the expression $x^2 - 4$ using the difference of squares method?
What is the result of factoring the expression $x^2 - 4$ using the difference of squares method?
In rational factoring, what would be the result of simplifying $\frac{2}{x^2} - \frac{3}{x}$?
In rational factoring, what would be the result of simplifying $\frac{2}{x^2} - \frac{3}{x}$?
How is the technique of factoring by taking out a common factor applied to factorize $12xy - 18y$?
How is the technique of factoring by taking out a common factor applied to factorize $12xy - 18y$?
If given the expression $a^3 - b^3$, what is the factored form according to the difference of cubes method?
If given the expression $a^3 - b^3$, what is the factored form according to the difference of cubes method?
Study Notes
Mastering Factorisation: Common Techniques Revealed
Factorisation, the process of breaking down an expression into its constituent parts, is a fundamental concept in algebra. Much like breaking down a complex puzzle into smaller, solvable pieces, factorisation offers a way to simplify expressions and solve equations. In this article, we'll explore some common factorisation techniques to empower your algebraic problem-solving skills.
Factoring by Grouping
Factorisation by grouping is a method that allows us to rearrange terms in an expression, factoring out a common term from each group. Here's an example:
[ 3x(x + 1) + 5(x + 1) = (3x + 5)(x + 1) ]
In this case, we factored out the common term ((x + 1)) from each group.
Difference of Squares
The difference of squares factorisation is a technique to find a factored form of an expression containing the difference of two squares, e.g., (a^2 - b^2). The general form is:
[ (a + b)(a - b) ]
For example:
[ x^2 - 9 = (x + 3)(x - 3) ]
Sum of Cubes
The sum of cubes factorisation is a technique to find a factored form of an expression containing the sum of two cubes, e.g., (a^3 + b^3). The general form is:
[ (a + b)^2 - a^2 - b^2 ]
Expanding, we get:
[ a^2 + 2ab + b^2 - a^2 - b^2 = 2ab ]
[ \therefore (a + b)^2 - a^2 - b^2 = 2ab ]
For example:
[ x^3 + 8 = (x + 2)(x^2 - 2x + 4) ]
Difference of Cubes
The difference of cubes factorisation is a technique to find a factored form of an expression containing the difference of two cubes, e.g., (a^3 - b^3). The general form is:
[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 ]
For example:
[ x^3 - 27 = (x - 3)(x^2 + 3x + 9) ]
Factoring by Taking Out a Common Factor
This method involves factoring out a common term from a single term or a product of terms. For example:
[ \begin{align} 2x + 6 &= 2(x + 3) \ 3ab - 9a &= 3a(b - 3) \end{align} ]
Rational Factoring
Rational factoring, also known as factoring by difference of quotients, is used when we have an expression of the form:
[ \frac{a}{b} - \frac{c}{d} = \frac{a \times d - b \times c}{b \times d} ]
For example:
[ \frac{x}{x + 1} - \frac{1}{x + 1} = \frac{x - 1}{(x + 1)^2} ]
Conclusion
Factorisation is a powerful tool in algebra that can be used to simplify expressions and solve equations. By mastering the techniques described above, you'll be well-equipped to tackle a wide range of algebraic problems.
Remember, this is just a brief overview of the most common factorisation techniques. As you deepen your understanding of algebra, you'll encounter more advanced techniques and applications. Keep exploring, and happy factoring!
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Description
Explore common factorisation techniques such as Factoring by Grouping, Difference of Squares, Sum of Cubes, Difference of Cubes, Factoring by Taking Out a Common Factor, and Rational Factoring. Improve your algebraic problem-solving skills with these powerful methods.
