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A short class with Sarah and her students. The other ways are under More.
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Today, we are going to explore an interesting way to factor expressions called 'factorisation by regrouping'. Can anyone explain what they think that means?
Is it like finding common factors in an expression?
Yes, exactly! It's about reorganizing terms to find those common factors. For example, in the expression 2xy + 2y + 3x + 3, if we group the first two and the last two terms, we can see something interesting.
What do you see when you group them?
The first two terms can be factored as 2y(x + 1) and the last two as 3(x + 1). So we can rewrite the whole expression as (x + 1)(2y + 3).
I see! That helps a lot! So we rearranged to find common factors.
Exactly! Let's remember: 'Group to find.' If you can identify groups that can simplify an expression, you'll unveil common factors!
Consider the expression 6xy - 4y + 6 - 9x. Can anyone identify potential groups here?
Maybe we can group 6xy with -9x and -4y with 6?
Great observation! Let's try that. If we rearrange it as 6xy - 9x - 4y + 6, we can factor the first group as 3(3y - 2) and the second group as -1(3y - 2).
So we can factor out (3y - 2) completely!
Correct! The final result is (3y - 2)(2x - 1). Excellent work, everyone! Remember, practicing the regrouping technique will make you much more adept at factorising tricky expressions.
To wrap up, factorisation by regrouping is a powerful tool. Are there any questions about the method?
Can this method be used on any polynomial?
Great question! While it can be very effective, it's not always guaranteed. Sometimes, the arrangement might not lead to a simple factorization. This is where practice matters!
So we just have to look carefully at the terms?
Exactly! Keep an eye out for those groupable terms. Always remember — 'look, regroup, factor!' Let's practice some more examples together.
A method to factor expressions by reorganizing them into groups that reveal common factors.
Look for similarities between terms to group potential factors.
The technique involves rearranging terms to facilitate easy grouping.
To group and to factor is quite a fun game, rearranging the terms will lead you to fame!
Once there was a wise math wizard who loved to solve puzzles. One day he faced a tricky expression, but he rearranged the numbers, grouping them cleverly until they revealed hidden treasures—common factors.
RAP - Rearrange, Analyze, and Factor.
GFG - Group Factors Gradually.
Term
What is regrouping?
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Term
How do you find common factors?
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The process of rearranging terms in an expression to form manageable groups that reveal common factors.
A number or term that divides two or more numbers or terms evenly.
The process of breaking down an expression into its constituent factors.
Factors that cannot be further broken down into simpler components.
This section teaches the method of factorisation by regrouping terms in algebraic expressions, highlighting the process through specific examples.
Factorisation by regrouping involves rearranging terms in an expression to reveal common factors, enabling the expression to be rewritten as a product of simpler expressions. The section illustrates this method with examples such as 2xy + 2y + 3x + 3, emphasizing the importance of recognizing groupable terms.
In this section, we explore the method of factorisation by regrouping terms, which is particularly useful in algebraic expressions where no single common factor is apparent. The process begins by identifying terms that can be grouped based on shared factors. For example, in the expression 2xy + 2y + 3x + 3, we can observe that the first two terms share 2y as a common factor, while the last two terms share 3. This allows us to factor each group individually:
This results in the expression being rewritten as (x + 1)(2y + 3). The significance of this method lies in its utility for expressions where obvious single factors do not exist, enabling a clearer pathway to factorisation. Furthermore, the technique encourages a deeper understanding of the relationship between terms and the common factors they may contain. Through multiple examples and explanations, students can develop skills to recognise how to rearrange terms effectively for factorisation.
Solution:
Step 1
Check if there is a common factor among all terms. There is none.
Step 2
Think of grouping. Notice that the first two terms have a common factor of 5y:
5xy - 10y = 5y(x - 2)
What about the last two terms? Observe them. If you change their order to -5x + 15, it will come out:
-5x + 15 = -5(x - 3)
Step 3
Putting (a) and (b) together,
5xy - 10y + 15 - 5x = 5y(x - 2) - 5(x - 3) = 5y(x - 2) - 5(x - 3)
The factors of (5xy - 10y + 15 - 5x) are (5y - 5)(x - 2)