document.write( "Question 175242: 18z+45+z^(2 )\r
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Algebra.Com's Answer #130319 by jim_thompson5910(35256) $\"\"$ $\"About$

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$\"18z%2B45%2Bz%5E2\"$ Start with the given polynomial\r
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\n" ); document.write( "\n" ); document.write( " $\"z%5E2%2B18z%2B45\"$ Rearrange the terms in descending order.\r
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"z%5E2%2B18z%2B45\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"18\"$ , and the last term is $\"45\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"45\"$ to get $\"%281%29%2845%29=45\"$ .

Now the question is: what two whole numbers multiply to $\"45\"$ (the previous product) and add to the second coefficient $\"18\"$ ?

To find these two numbers, we need to list all of the factors of $\"45\"$ (the previous product).

Factors of $\"45\"$ :

1,3,5,9,15,45

-1,-3,-5,-9,-15,-45

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

These factors pair up and multiply to $\"45\"$ .

1*45 = 45
3*15 = 45
5*9 = 45
(-1)*(-45) = 45
(-3)*(-15) = 45
(-5)*(-9) = 45

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"18\"$ :

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First Number	Second Number	Sum
1	45	1+45=46
3	15	3+15=18
5	9	5+9=14
-1	-45	-1+(-45)=-46
-3	-15	-3+(-15)=-18
-5	-9	-5+(-9)=-14

From the table, we can see that the two numbers $\"3\"$ and $\"15\"$ add to $\"18\"$ (the middle coefficient).

So the two numbers $\"3\"$ and $\"15\"$ both multiply to $\"45\"$ and add to $\"18\"$

Now replace the middle term $\"18z\"$ with $\"3z%2B15z\"$ . Remember, $\"3\"$ and $\"15\"$ add to $\"18\"$ . So this shows us that $\"3z%2B15z=18z\"$ .

$\"z%5E2%2Bhighlight%283z%2B15z%29%2B45\"$ Replace the second term $\"18z\"$ with $\"3z%2B15z\"$ .

$\"%28z%5E2%2B3z%29%2B%2815z%2B45%29\"$ Group the terms into two pairs.

$\"z%28z%2B3%29%2B%2815z%2B45%29\"$ Factor out the GCF $\"z\"$ from the first group.

$\"z%28z%2B3%29%2B15%28z%2B3%29\"$ Factor out $\"15\"$ 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.

$\"%28z%2B15%29%28z%2B3%29\"$ Combine like terms. Or factor out the common term $\"z%2B3\"$

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Answer:

So $\"z%5E2%2B18%2Az%2B45\"$ factors to $\"%28z%2B15%29%28z%2B3%29\"$ .

In other words, $\"z%5E2%2B18%2Az%2B45=%28z%2B15%29%28z%2B3%29\"$ .

Note: you can check the answer by expanding $\"%28z%2B15%29%28z%2B3%29\"$ to get $\"z%5E2%2B18%2Az%2B45\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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