What are Factors?

There are two types of factors:

• Numerical Factors: Numbers that divide evenly into another number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
• Algebraic Factors: Expressions that divide evenly into another expression.

Factorisation is the process of breaking down an expression into its factors (like writing it as a multiplication problem). Think of it as the reverse of expanding expressions using the distributive law.

For example, let’s factor the number 12. 12 can be written as 2 * 6 or 3 * 4 or 2 * 2 * 3. Another example is: What are the common factors of 15 and 25? 1, 5

To factorise, find the Greatest Common Factor (GCF). The GCF is the largest number that is a factor of two or more numbers or terms.

For example: Find the GCF of 12 and 18. Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. The GCF of 12 and 18 is 6.

Now, the GCF is important when you’re factorising algebraic expressions. Here’s how to factorise algebraic expressions:

1. Find the GCF of the numerical coefficients of all terms.
2. Write the GCF outside a set of parentheses.
3. Inside the parentheses, write the remaining factors of each term after dividing by the GCF.

Here’s an example: Factorize 6x + 9y. The GCF of 6 and 9 is 3. Write 3 outside the parentheses: 3( ). Divide each term by 3: 6x / 3 = 2x and 9y / 3 = 3y. Write the results inside the parentheses: 3(2x + 3y).

Easy right?

Let’s do some practice questions (Watch the video for detailed workings):

1. Factorize 4x + 6y.
2. Factorize 12p + 18q - 6r.
3. Factorize 9m - 15n + 21p.

Solutions:

1. 2(2x + 3y)
2. 6(2p + 3q - r)
3. 3(3m - 5n + 7p)

Problem Solving with Algorithms and Error Detection

Algorithms are like recipes – sets of instructions that describe how to solve a problem or complete a task. In math, algorithms can be used to perform calculations, simplify expressions, or solve equations.

A simple instruction sheet for making a cup of instant noodles could be considered an algorithm if there are step-by-step instructions in order and a defined outcome, and if you repeat the steps correctly, you should get the same results each time.

In math, algorithms are similar. They are specific sets of instructions used to solve problems. For example, the steps you follow when simplifying an algebraic expression or factorizing it can be considered an algorithm. Just like making instant noodles, if you follow the correct steps in the correct order, you'll arrive at the correct answer.

But we know that things don’t always go right – and sometimes, you get an outcome that is unexpected. Then, it’s important to learn how to identify and correct errors.

When working with algebraic expressions, common mistakes include:

• Incorrectly distributing a negative sign.
• Forgetting to divide all terms by the GCF.
• Combining terms that are not like terms.

To check your work (and find errors), you can:

• Substitute values: Choose simple values for the variables and substitute them into both the original expression and the simplified/factorized expression. The results should be the same.
• Work backward: If you've factorised an expression, try expanding it back to see if you get the original expression.

Example: Let's say a student factorized 8x + 12y as 2(4x + 12y).

Let’s detect the error by using substitution: Let x = 1 and y = 1. Original: 8(1) + 12(1) = 8 + 12 = 20. Factorized: 2(4(1) + 12(1)) = 2(4 + 12) = 2(16) = 32. The results are different!

So, what’s the error? The student did not take out the greatest common factor (which is 4, not 2).

Correction: The correct factorisation is 4(2x + 3y).

Solving Problems – Factorisation, Algorithms and Error Detection

Let’s do some more trickier questions – remember, in the exam, it’s unlikely you’ll receive a question that asks you to factorise a direct expression, you’ll likely need to comprehend the question and then come up with the expression and work with it!

Question 1: The Bake Sale Balance Sheet

A school is organising a bake sale to raise funds. They are selling two types of baked goods: cookies and brownies.

• The profit from each cookie sold is $5.
• The profit from each brownie sold is $10.

On the first day, they sold 'x' number of cookies and 'y' number of brownies. On the second day, they sold 3 fewer cookies than the first day and 5 more brownies than the first day.

A student is trying to factorise the expression for the total profit from both days. The student's algorithm for factorising is as follows:

5X +10 y + (5x-3) + (5+10y) =
1. Calculate the profit from each day.
2. Add the profits together.
3. Factor out a 5.
4. The student's factorized expression is 5(2x + 3y + 4).

(a) Is the student's algorithm and final factorised expression correct for the total profit from both days?

(b) If the student's algorithm or expression is incorrect, identify the error(s) and provide the correct algorithm and factorized expression.

(c) Using the correct expression, if x=20 and y=10, was the profit on day 1 more than the profit on day 2?

Solution:

Part A

• Calculate profit from Day 1: 5x + 10y
• Calculate profit from Day 2: 5(x - 3) + 10(y + 5) = 5x - 15 + 10y + 50 = 5x + 10y + 35
• Add the profits together: (5x + 10y) + (5x + 10y + 35) = 10x + 20y + 35
• Factor out a 5: 5(2x + 4y + 7)
• Compare with the student's expression: The student's expression 5(2x + 3y + 4) is incorrect.

Part B

Error: The student's algorithm is partially incorrect. While the steps are correct, they made an error in calculating the profit on day 2 by incorrectly simplifying 5(x-3) + 10(y+5). Therefore, the student's final factorised expression is incorrect.

Correct algorithm:

• Calculate the profit from each day separately, in terms of x and y.
• Add the expressions for the profit from each day to find the total profit.
• Factor out the greatest common factor (which is 5 in this case).
• Correct factorized expression: 5(2x + 4y + 7)

Part C

• Substitute x=20 and y=10 into the profit expressions for each day:
• Day 1: 5(20) + 10(10) = 100 + 100 = $200
• Day 2: 5(20) + 10(10) + 35 = 100 + 100 + 35 = $235
• Compare the profits: Day 2 profit ($235) is more than Day 1 profit ($200).

These questions are a bit more complicated and the way you can work through them is taking the time now, slowly, to understand the process because the more familiar you are with the calculations, the easier it will be to problem solve in the exam!

Practice time!

Now, it's your turn to practice.

The questions in this checkpoint are provided to give you an introduction to possible questions you may see in your exam. Don't worry too much as you'll continue to build your skills throughout the course.

Click on the button below and start your practice questions. We recommend doing untimed mode first, and then, when you're ready, do timed mode.

Every question has a suggested solutions videos after you complete the question. This video explains to you the steps to take to answer the question and provides tips and tricks.

Once you're done with the practice questions, move on to the next checkpoint.

Now, let’s get started on your practice questions.

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