Question

Factor the following trinomial.
w^18-9w^9y^5+14y^10

Answer 1

\[w^{18}-9w^9y^5+14y^{10}\]

Answer 2

what are the coefficients of the first and last terms?

Answer 3

idk w^18 and 14y^10?

Answer 4

the coefficients are just the numbers in front...
if there is no coefficient visible, it is 1: \(w^{18} = 1w^{18}\)

Answer 5

Multiply the coefficients of the first and last terms. Next, find the factors of that number which when added together, give you the coefficient (including the sign!) of the middle term.

Answer 6

im so confused

Answer 7

what is the coefficient of \(w^{18}\)?

Answer 8

\[1w ^{18}\]?

Answer 9

Again, the coefficient is the number in front of the variable(s). If there is no number in front of the variable(s), the coefficient is 1.

What is the coefficient of \(w^{18}\)?

Answer 10

1

Answer 11

very good. What is the coefficient of \(14y^{10}\)?

Answer 12

okay, it's 14. We multiply the coefficients of the leading and trailing terms:

1*14 = 14

Now, we find a pair of factors of that product which add up to the coefficient of the middle term, which is -9 in this problem. We need a positive product and a negative sum, so we'll need to use two negative numbers (two negative numbers multiply to give a positive number, but sum to a negative number).

-1*-14 = 14, -1 + -14 = -15 no good
-2*-7 = 14, -2 + -7 = -9 that's what we want

So we take our two factors and use them to "split" the middle term:

\[-9w^9y^5= -7w^9y^5 - 2w^9y^5\]
Our factoring is complete and correct.
giving us
\[w^{18}-7w^9y^5-2w^9y^5+14y^{10}\]We put parentheses around each pair of terms:
\[(w^{18}-7w^9y^5)+(-2w^9y^5+14y^{10})\]Note that I put the parentheses around the negative sign — doing it this way reduces the risk of a common error.

Next, we factor out the common factors in each group:
\[w^9(w^9-7y^5) + (-2y^5)(w^9-7y^5)\]
Look at that — each group has a common factor with the other group!
\[(w^9-7y^5)(w^9+(-2y^5))\]\[(w^9-7y^5)(w^9-2y^5)\]

Finally, test our factoring by multiplying it out and making sure we get what we started with.
\[(w^9-7y^5)(w^9-2y^5) = w^9*w^9+w^9(-2y^5) -7y^5*w^9 -7y^5(-2y^5)\]\[\qquad=w^{18} -2w^9y^5-7w^9y^5+14y^{10}\]\[\qquad=w^{18}-9w^9y^5+14y^{10}\checkmark\]

Answer 13

Hmm, not sure how the "Our factoring is complete and correct." line got up in the middle; it should have been at the end!