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Expanding

Review:

  • In grade 9, you should have lerarned about the distrubutive property

  • ex. 2(3-x)

             6-2x

             (multiply 2 by both 3 AND x)

 

Now lets add on to that:

  • with two brackets beside each other (and terms inside the two brackets-this is called factored form), you would multiply everything in the first bracket by everything in the second bracket

  • ex. (x+1)(x+2)

             -x multiplied by x, then x multipled by 2

             -1 multiplied by x, then 1 multipled by 2

            = x +2x+x+2

             -when you're asked to expand, you usually have to simplify aswell (so, collect the "like terms")

             =x +3x+2

2

2

Common Factoring

Factors are numbers that are multiplied together to make your product.

Common factoring is usually just the first step to take , then you may be dealing with complex trinomials, difference of squares etc.

Note: When you factor, you must be able to factor eveything in the given equation; without getting decimlas or creating negatives

  • ex. (2x+6)

             =2(x+6)

           -you can factor out a 2

           -you can't factor anything larger than 2, because then the 2 would become a decimal

             ex. =4(0.5x=6)---THIS IS WRONG

           -you also can't factor out an (x) because not all terms have (x) in them

  • ex. 2x -4x

             =2x (x-2)

            -you can factor out a 2x

            -Note: When you have an x , you just write x (3-2=1). When you have an x , you don't write the x (2-2=0).

            -you can't factor out anything more than a x , because then, 4(x ) would become a negative

3

2

2

2

2

2

1

0

Grouping

This type of factoring usually conatins 4 terms

 

Steps to take:

1. Put "like terms" together (in brackets)-you should have it in the format of factored form

2. Common factor

3. What's left in the brackets should be the same terms, cancel those terms out by dividing them by themselves

4. What's left (what you common factored out) and what was left in the brackets (in step 3) is the answer

5. Simplify if you can

  •  

Simple Trinomials

Simple trinomials have 3 terms and there is a co-efficient of 1 infront of the x

 

Steps to take:

1. Find 2 numbers that can be multiplied together to give you the last term

2. Check if those 2 same numbers can be added together to give you the middle term

3. If they can't, try another 2 numbers

4. If they can, then the answer would be (x +/- one of those numbers) and (x+/- the other number)

  • It's negative/positive based on whether the numbers are positive/negative.

  • It' (x +/- the number) because the first term will always be x , and x multiplied by x would give you x .

..

2

2

2

Complex Trinomials

Complex trinomials have 3 terms and this time, there is a co-efficient larger than 1 infront of the x

 

Step to take:

1. Guess and check!!!

2. Write down multiplies of the first term and last term

3. Plug them in the format of factored form

  • The 2 multplies of the first term, go first in both brackets. Then the 2 multiples of the last term, go second in both brackets.

4. Check by expanding (you can check all your answers, in all types of factoring by expanding)

Perfect Squares

Perfect squares also have 3 terms. However, it's only a perfect square if you can square-root the first and last term and get a whole number.

 

Steps to take:

1. Square-root the first and last term

  • Whether the last term is a postive or negative, depends on whether the middle term is positive or negative.

2. You would write this answer in factored form...

  • However both of the brackets are the exact same. So you would just write the answer in a bracket and then put the squared sign outside of it.

3. You can also check this by expanding. (or you can multiply both square-rooted numbers by each other and multiply them by 2)

Difference of Squares

2

Difference of squares has only two terms. The last term is a negative and both terms can be sqaure-rooted to give you a whole number.

 

Steps to take:

1. Sqaure-root both terms

2. When you square-root the last term, one should be a positive and one should be a negative

  • This is so when you expand, the middle term is cancelled out and you get only two terms, like in the original equation.

3. Write your answer in factored form

  • When you sqaure-root the first term, it goes first in both brackets. Then, when you square-root the last term, a negative version of that would go into one bracket and a positive version of that would go into the other.

4. Check your answer by expanding.

Connection: You use these techniques of factoring in standard form, so you can change it to factored form. In factored form you can find the zeros (by setting the x to zero), you can also find the vertex (find the x-value by adding your zeros and dividing the answer by 2, find the y-value by plugging in the x-value in the original equation).

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