1. ## polynomial expansion

one of the terms in the expansion $\displaystyle (x+3)^7$ is $\displaystyle kx^5$. determin the value of k.

2. ## finding the pattern

$\displaystyle (x+y)^{1}=x+y$
$\displaystyle (x+y)^{2}=x^{2}+2yx+y^{2}$
$\displaystyle (x+y)^{3}=x^{3}+3yx^{2}+3y^{2}x+y^{3}$
$\displaystyle (x+y)^{4}=x^{4}+4yx^{3}+6y^{2}x^{2}+4y^{3}x+y^{4}$

The pattern of coefficients is like pascals triangle, the outside is always 1's and you add the two coefficeints in the line directly above to get the inside ones:

...1 1
..1 2 1
.1 3 3 1
1 4 6 4 1

As for the 3 (or y value see above) when you are talking about a power of 7 the power of the y will always be 7 - power of x in that part of the polynomial.

Try to figure it out and post the solution if you can't get it or if you want me to check it I will give the answer.

3. ## i think i got it

the next line would start with $\displaystyle x^5$ so the coefficient or the k value would be 1.

4. yes but the next line is to the power of 5, you are asking about power of 7. so the k you are after is actually different.

5. ## think i got it

so then if it was like the trangle it would be like 7 chose 2 or 21

6. 21 is correct but that is not k.

you also need the 3 or y component.

$\displaystyle 21*y^{2}*x^5$
or
$\displaystyle 21*3^{2}*x^5$