Let’s start by understanding factors.
“Factors” in the simplest terms can be thought of as the numbers we can multiply together to get another number.
Examples of Factors:
- 2 × 2 = 4, in this case 1 and 2 are factors of 4
- 2 × 3 = 6, here 1, 2 and 3 are all factors of 6
- 3 x 4 = 12, here the factors of 12 are 1, 12, 2, 6, 3 and 4.
The second way to think about factors of a number is to remember the rule that the number should be evenly divided by its factors.
Therefore, factoring is all about finding the numbers, and/or algebraic expressions that can be multiplied together to get a final number or a final equation.
The knowledge of factoring is important to students starting middle grade as not only it can help students solve algebraic equations but also help in finding solutions to quadratic equations or complex polynomial problems in higher grades.
The art of factoring also comes in handy when taking the SAT exam as the skill of factoring quickly can help you narrow down on your multiple-choice questions through the method of elimination.
Did you know that variable expression can be factored in?
Yes, you heard it right. It is very much possible to factor variables with numeric coefficients. Let’s take a few examples to understand this clearly:
We will take the same example as above and add an “x” to 12 making it 12x. Now when you try to factor this remember all the factors, we found on the list above and write them:
3(4x) or 2(6x) and now further breaking the 4 & 6, you can factor it as – 3(2(2x) or 2(3(2x)
Now we can progress a step further and start to deal with factoring the algebraic equations.
In this case, we will be using the distributive property, also referred to as the distributive law of multiplication and division.
Let’s take the same example we took above of 12x and add 6 to it. So, the expressions end up looking something like this: 12x +6
So now applying the distributive property a(b+c), we can take our equations simplified to
We can try one more example keeping 12x and adding 8 to it. So the expression becomes 12x+8.
Now let’s apply the distributive property once again and try to simplify the expression.
We are sure by now you feel confident dealing with simple factoring & algebraic expressions.
Are you ready to take on the challenge of factoring quadratic equations?
Let’s first understand, what is a quadratic equation in algebra?
In algebra, a quadratic equation is an equation that is arranged in the following form:
a, b, c = known numbers, where a ≠ 0
x = the unknown
Solving quadratic equations can easily get tricky so ensure that you build a strong foundation about this concept. So let’s explore all the different ways that one can solve a quadratic equation in algebra.
- A quadratic equation can be solved through factoring
- Can also be solved using the square roots
- One can solve by completing the square
- And by using the quadratic formula
Although there are four ways, as shown above, of solving the quadratic equation. For today, we are going to only focus on the first method that is solving the quadratic equation by factoring.
Let’s take an example to understand usage of factoring –
5x^2 + 7x – 9 = 4x^2 + x – 18 (equation)
Step 1 (5x^2 + 7x – 9)-( 4x^2 + x – 18 ) = 0
Step 2 (5x^2 – 4x^2 ) + (7x -x) +( – 9 + 18 ) = 0
Step 3 (x^2) + (6x) – ( 9 ) = 0
Now let’s take a^2 -b^2 type of an equation and solve:
The equation is 9x^2 – 4y^2
Let’s use – a^2 -b^2 = (a+b) (a – b)
The simplest form of a = 3x and b = 2y
The solution is (3x + 2y)(3x – 2y).
How about trying an a^3 -b^3 problem?
8x^3 – 27y^3
Let’s use – a^3 -b^3 = ( a-b) (a^2 + ab + b^2)
In this case the simplest form of a =2x and b =3y
Factoring the solution to (2x – 3y)(2(2x^2) + ((2x)(3y)) + 3(3y^2))
Let’s take one more example to understand usage of factoring but this time we will use factoring by inspection –
3x^2 – 8x + 4 = 0
Now adding -2 to both blank spaces gives the correct answer. -2 × 3x = -6x and -2 × x = -2x. -6x and -2x add to -8x. -2 × -2 = 4, so we can see that the factored terms in parentheses multiply to become the original equation.
(3x) (x) – 8x + (-2) (-2) = 0
(3x) x (-2) = -6x
(x) x (-2) = -2x
(3x-2)(x-2) = 0
If you feel confident about factoring algebraic & quadratic equations, then take this challenge and try to solve these questions in under 1 minute each:
- How do I factorize 4x^3 + 8x?
- How do I factorize x^2 + 6x + 9?
- What are the factors of 2x^2 – 7x-6 = 0?
- How would your factor be -24x+4x^2?
- Try factoring x2 + 5x + 6 = 0.
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