3(x-5) +4(x-5). Notice the group formed, x- 5, occurs twice.
That is our first goal (step).
Try this one: 4x-8 + 3xy- 6y. This will become 4(x-2) + 3y(x-2).
I did this by common factoring the 4 from the 4x - 8
and then common factoring the 3y from the 3xy - 6y.t
Notice that the group (x-2) occurs twice.
I, of course, was careful in creating the 4 terms of the polynomial!
I wanted there to be an expression that was repeated.
Try these 2 problems.
1. 5x-20+ 2x-8
2. xy-7y + x -7
Hint: for the last 2 terms common factor a 1 from each!
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Example A | Example B |
|
5(x-4) + 2(x-4) |
|
y(x-7) + 1(x-7) |
4x- 8 + 3xy- 6y = |
Notice the 2 expressions inside |
4(x-2) + 3y(x-2) |
the parenthesis are EQUAL. |
|
I want to temporarily |
|
call the (x-2) BIG P |
4P + 3yP |
Replace each (x-2) with P. |
|
Now this is an easy problem to factor. As we did in Lesson 3.1. Do so. |
or P(4 +3y) |
The 2 terms have a BIG P in common. |
= (x - 2)(4 + 3y) |
Now replace the P with its equal (x-2)
|
THUS 4x - 8 + 3xy - 6y factors to (x-2)(4+3y).
Be sure you see that we had exactly the same expression in each parenthesis.
We temporarily replace all of the identical parenthesis with P
and then see we can factor with common factoring of the P.
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Example A | Example B | |
5x -20 + 2x -8 |
|
yx-7y + x - 7 |
5(x-4) + 2(x-4) |
|
y(x-7) + 1(x-7) sometimes we have to use a 1 for the common factor. |
5P + 2P |
|
yP + 1P |
P(5 + 2) |
|
P(y+1) |
P7 = (x-4)7 |
|
(x -7)(y+1) |
Remember this BIG P Principle of Factoring.
It would be a good idea for you to copy these to your notebook. We will use this again.
4x+8 -3xy- 6yLOOK HERE FIRST. Do you see the signs are not the same for our groups? The first 2 terms have positives and the last two have negatives. When the signs of the first 2 terms are not the same as the signs of the last two terms then that is a BIG SIGNAL to you. You must factor out a negative from the last two terms and that will change the last 2 signs to the opposite of what they are. Read this again. Then read the box to the right starting at the top.
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Factor a 4 from the 4x+8 and then factor a -3y from the -3xy- 6y. I did that not because they are both negative, but because they are not the same as the signs of the first 2 terms, 4x+8. THEY ARE OPPOSITES! 4x+8= 4(x + 2) and -3xy - 6y = -3y(x + 2) We have the same terms in both parenthesis. YEA! Again, if you do not understand this then go back to common factoring.
When the signs of the first two terms are not the same as the signs of the last two terms then use a negative common factor, see the -3y in this problem.
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4(x+2) - 3y(x+2) Notice the 2 expressions inside the parenthesis are EQUAL. |
See the - 3y. When I factor it out of the -3xy- 6y it will make both of these positive. Let's check that part. -3xy-6y = -3y(x+2) Right? Multiply the -3y(x+2) back out and you should get the -3xy -6y. Be sure you see this. Notice the 2 expressions inside the parentheses are EQUAL. In this problem they are both x+2. They MUST be equal or something is wrong or it is not this type of factoring problem!!! There are other kinds!! Be sure you have not made a mistake.
|
|
I want to temporarily call (x+2) a BIG P. |
4P - 3yP |
Replace each (x+2) with P. |
|
Now this is an easy problem to factor, that is, if you know common factoring. |
= P(4 - 3y) |
The 2 terms have a BIG P in common. |
becomes (x + 2)(4 - 3y) |
Now replace the P with its equal (x+2) |
Now remember if the signs of the first two terms are not the same
(same order also) as the last two terms then that is your BIG SIGNAL to
common factor out a negative and change the signs of the last two terms.
Another similar, but different!, Example:
Are the signs of the first two terms similar to the last two terms in this one? 5xv - 20v - 2x + 8
No. The first two terms: positive and then a negative. The last two terms: negative and then a positive.
This is our BIG SIGNAL to factor out a negative from the last two terms and that will change their signs.
5xv - 20v - 2x + 8 = 5v(x - 4) - 2(x - 4) See the -2 factor. See the signs inside the parenthesis. The x is now positive and the 4 is negative. Just to check it, does -2(x - 4) = -2x + 8 ?
So here are the steps for you to write in your notebook: 5vx - 20v - 2x+ 8= 5v(x- 4) - 2(x- 4)= NOTE: x- 4 is inside both parenthesis 5vP-2P= P(5v -2) Replace the P. ( x- 4)(5v-2)
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x3+8x2- 3x2- 24x= ????= x2(x+8) - 3x(x+8) .
You finish then find answer below. No peeking. Write this one down and factor it completely. |
Do this problem then look at the answer below. What do the first two terms have in common??? x2 What do the last two terms have in common??? - 3x
x3+ 8x2- 3x2- 24x = x2(x+8) - 3x(x+8) Think: x2P- 3xP = P(x2- 3x) (x+8)(x2-3x) = But we need to look more at the x2-3x. (x+8)x(x-3) or x(x+8)(x - 3)
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Do Assignment 3.2a. Look in the Section 1.
Can you common factor when the exponent is a variable too!
x2n- xn+1+ xn do you see all have xn in common.Factor xn out!
BUT first do you recall that when multiplying we add the exponents.
Thus xn times xn is x2n ( I added n+n).
So what is xn times x. What are the exponents?n and 1 so we get xn+1. |
x2n- xn+1+ xn You may want to think xnxn - xnx1+ 1xn .
factors to xn(xn- x1+ 1) OR xn(xn- x + 1) |
You will need the above in coming lessons!