Thinking about degrees of relation in real life

Last learning activity, you looked at graphs of relations to determine if they were linear, quadratic, or neither.

Factoring expressions

Quadratic expressions can be represented in both expanded and factored form. In the last learning activity you expanded expressions from factored form to expanded form.

The opposite of expanding is factoring. In this learning activity, you will learn how to factor expressions.

A factor is a number that is multiplied by another number to give a specific product. In other words, a factor divides evenly into a product. When a number is factored, it is represented as a product of two or more factors.

• For instance, if you want to build a shed so that the area of the floor is $100m^2$, you can choose from a variety of possible dimensions. A $100m^2$ shed could be $10m\times 10m$, or $2m\times 50m$ or $4m\times 25m$ or $1m\times 100m$ These dimensions represent factors of $100$.

Using a real life example to explain factoring

The greatest common factor (GCF)

Factoring type 1

We should always identify a common factor before attempting any other type of factoring. You can factor out any common factor in an expression or equation. Usually it is best to factor out the GCF.

A factor is a number that evenly divides into another. For instance $3x$ has factors:

• 1 and $3x$
• $1x$ and 3

Using the example: factor ${6x}^4+12x^{3}y-{3x}^2{y}^2$

Another approach for finding the GCF

Here's another way to aproach factoring out the GCF:

Let's say you want to find the GCF of ${6x}^4+12x^{3}y-{3x}^2{y}^2$.

Imagine each term represented as the product of its factors.

Here is how you would circle all the factors that are the same in each row.

$6x^4+12x^{3}y-3x^2{y}^2$

$6x^4:2\times \overline{)3}\times \overline{)x}\times \overline{)x}\times x\times x$

$12x^{3}y:2\times 2\times \overline{)3}\times \overline{)x}\times \overline{)x}\times x\times y$

$-3x^2{y}^2:-1\times \overline{)3}\times \overline{)x}\times \overline{)x}\times y\times y$

In this case they are $3$, $x$ and $x$.

The GCF is $3x^2$.

The factors of each term that are not part of the GCF will form the terms inside the brackets when you divide out $3x^2$.

${6x}^4+12x^{3}y-{3x}^2{y}^2$ = $3x^2(2x^2+4xy-y^2)$

Expanding and factoring

Expanding and factoring are inverse processes.

Expanding: the process of multiplying factors to state an expression in simplified form.

Factoring: the process of representing an algebraic expression in terms of its factors.

It is important you notice that not all expressions can be factored.

Think about this:

$3a(2a-5)=6a^2-15a$

Going from left to right, you are expanding; using the distributive property, the term $3a$ is multiplied through the other factor $2a-5$.

Going from right to left, you are factoring; the expression $6a^2-15a$ can be represented as the multiplication of two factors.

Self-check

As a self-directed learner, you will be reflecting on your learning process and checking your understanding in order plan for success. Make a note of your understanding of the success criteria from today’s activity.

Rate your understanding on a scale of five to one.

Five means “I have a thorough understanding.” One means “I am confused.”

Are you able to

Once you feel you are able to fulfill the success criteria, complete the following examples and confirm your solutions with the suggested answers.

Further practice factoring using the GCF

Factoring is a powerful tool for revealing important features of mathematical relations, like the vertex of a parabola. Factoring also can be used to make an algebraic equation easier to solve. Factoring out the GCF usually makes other types of factoring more straightforward because the terms inside the brackets will have smaller coefficients and less complicated variables.

If you don’t notice that there is a GCF in an expression until after carrying out other types of factoring you can divide it out then.

Using distributive properties review

Quadratic expressions can be expanded by using the distributive property and then simplified by collecting like terms.

For the product of a monomial and a binomial, the distributive property states that:

$a(b+c)=ab+c$

For the product of a binomial and a binomial, apply the distributive property states that twice:

$(a+b)(c+d)=a(c+d)+b(c+d)$

$=ac+ad+bc+bd$

Like terms have the same variables with the same exponents.

Three special multiplication patterns are:

$(a+b{)}^2=(a+b\left)\right(a+b)=(a^2+ab+ba+b^2)=a^2+2ab+b^2$

$(a-b{)}^2=(a-b\left)\right(a-b)=(a^2-ab-ba+b^2)=a^2-2ab+b^2$

$(a+b)(a-b)=(a^2-ab+ba-b^2)=a^2-b^2$

One way to factor a polynomial is to examine the GCF of its terms as one of its factors. For example, $6x^2+2x-4$ can be factored as $2(3x^2+x-2)$, since $2$ is the GCF of each term.