Algebra 1 > Notes > Yorkcounty Final > Yorkcounty > Greatest Common Factor

Greatest Common Factor

One of the skills you need to know to understand factoring is distribution: Ex: 4x (2x + 3y + 7) = 4x (2x) + 4x (3y) + 4x (7) =

8x2

+ 12xy

+ 28 x

Factoring is the opposite of distribution: Instead of distributing, you will pull out the biggest factor from all terms and make up a distribution problem: Factor out the greatest common number from all coefficients -you can do this on the TI-83 by pressing Math, Num, 9 and entering the numbers: gcd(16,20) for example Factor out the greatest exponent of all terms for each variable that is in all terms of the polynomial

Example 

Example 1: Factor the GCF from 6a2 + 12a + 30

Example 1: Solution First, we will factor a GCF from the polynomial coefficients The greatest common factor of 6, 12 and 30 is the biggest number which will divide into all three: 6 Next we will take the greatest exponent from the three variables attached to the coefficients. The 6 has an a2 attached, the 12 has an a, and the 30 has no variable. Since all terms have no variable, we will not be able to take out a variable term. Now we can make up a distribution problem, with the 6 taken out. 6 (a2 + 2a + 5) would be the correct answer because if we re-distribute the 6, we will get what we started with. The best answer is C).



Factor out the GCF from 8x2 - 12x + 16

Try This Solution: The greatest numerical factor for 8, 12, and 16 is 4. There is no greatest exponent since the last term, 16, has no variable. We can’t take out any x’s. 4 (2x2 -3x + 4) Checking your answer: When we re-distribute, we should get the expression in the problem. 4 (2x2 -3x + 4) 4(2x2) -4(3x) + 4(4) 8x2

-12x

+ 16

It Works!!



Factor out a GCF from x3 + 5x2 + x

x3 + 5x2 + x The coefficients of the expression are 1, 5 and 1: 1x3 + 5x2 + 1x The greatest common factor of 1,5,1 is one, so there is no factor to take out of the expression. Now the variables: we have powers of x attached to each coefficient. x3 + 5x2 + x has powers of x which are x3, x2, and x1. The lowest exponent (the most number of x’s any one coefficient has) is 1, so the most number of x’s we can factor out is 1 from each term. So, we can only factor out 1x1 or x. x(x2 +5x +1) is the distribution statement we make up. Check with distribution: x(x2 +5x +1) x(x2) + x(5x) +x(1) 3

2

Try This The formula for the total surface area of a rectangular prism is S = 2LW + 2LH + 2HW. Solve this equation for W.

You have to find a GCF of W to solve for W.

S= 2LW + 2LH -2HW: Since we are solving this equation for W, we want to isolate the W on the right and move all other terms to the left side. The only term on the right side that does not have a W in it is 2LH, so we will move that term to the left first to help get W isolated: S

=

2LW + 2LH + 2HW

S -2LH=

2LW + 2LH -2LH +2HW

subtract 2 LH on both sides S- 2LH =

2LW +2HW

simplify S - 2LH =

W (2L + 2H)

factor out W to help isolate it

S - 2LH =

W (2L + 2H) divide both sides by 2L - 2H

S - 2LH = (2L - 2H)

W (2L + 2H) (2L + 2H)

canceling on the right S - 2LH = (2L + 2H)

W

Try This 



rx = b + qx Solve for x

Solution rx = b + qx: you are solving for x and they aren’t both on the same side-- so move them both to the same side and move the term without x, b, to the other side. rx = b + qx rx - rx = b+ qx - rx 0

move x’s to right

= b + qx - rx

0 - b

= b - b + qx - rx

move “b” to left

-b

= qx - rx

-b

= x ( q - r)

factor out x

-b

= x (q - r)

divide both sides by q-m

q-r -b q- m

= = x

q-r simplify for answer