SOLUTION: Hi all, just wondering if someone could show me how to factorise the following. Steps and explanations would be great. z^2-5z-14 Thanks -Nick.

Algebra ->  Distributive-associative-commutative-properties -> SOLUTION: Hi all, just wondering if someone could show me how to factorise the following. Steps and explanations would be great. z^2-5z-14 Thanks -Nick.      Log On


   

Question 200486: Hi all, just wondering if someone could show me how to factorise the following. Steps and explanations would be great.
z^2-5z-14
Thanks -Nick.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression z%5E2-5z-14, we can see that the first coefficient is 1, the second coefficient is -5, and the last term is -14.


Now multiply the first coefficient 1 by the last term -14 to get %281%29%28-14%29=-14.


Now the question is: what two whole numbers multiply to -14 (the previous product) and add to the second coefficient -5?


To find these two numbers, we need to list all of the factors of -14 (the previous product).


Factors of -14:
1,2,7,14
-1,-2,-7,-14


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to -14.
1*(-14)
2*(-7)
(-1)*(14)
(-2)*(7)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -5:


First NumberSecond NumberSum
1-141+(-14)=-13
2-72+(-7)=-5
-114-1+14=13
-27-2+7=5



From the table, we can see that the two numbers 2 and -7 add to -5 (the middle coefficient).


So the two numbers 2 and -7 both multiply to -14 and add to -5


Now replace the middle term -5z with 2z-7z. Remember, 2 and -7 add to -5. So this shows us that 2z-7z=-5z.


z%5E2%2Bhighlight%282z-7z%29-14 Replace the second term -5z with 2z-7z.


%28z%5E2%2B2z%29%2B%28-7z-14%29 Group the terms into two pairs.


z%28z%2B2%29%2B%28-7z-14%29 Factor out the GCF z from the first group.


z%28z%2B2%29-7%28z%2B2%29 Factor out 7 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


%28z-7%29%28z%2B2%29 Combine like terms. Or factor out the common term z%2B2

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Answer:


So z%5E2-5z-14 factors to %28z-7%29%28z%2B2%29.


Note: you can check the answer by FOILing %28z-7%29%28z%2B2%29 to get z%5E2-5z-14 or by graphing the original expression and the answer (the two graphs should be identical).