We need to find the coefficient of x12 in the expansion of (x2+2x+2)8. Step 1: General Term Formula Each term in the expansion is made by choosing P1 times x2,P2 times 2x, and P3 times 2 from the 8 factors, so P1+P2+P3=8. The general term is: ‌
8!
P1!P2!P3!
(x2)P1(2x)P2(2)P3 Step 2: Power of x The power of x in each term is 2P1 (from x2 ) plus P2 (from 2x ), for a total of 2P1+P2. Step 3: Find Values for x12 We want 2P1+P2=12 and P1+P2+P3=8. Possible solutions are:
P1
P2
P3
6
0
2
5
2
1
4
4
0
Step 4: Write Each Term's Coefficient For each solution, we substitute into the formula: When P1=6,P2=0,P3=2 : Coefficient: ‌
8!
6!0!2!
⋅20⋅20⋅22 When P1=5,P2=2,P3=1 : Coefficient: ‌
8!
5!2!1!
⋅22⋅21 When P1=4,P2=4,P3=0 : Coefficient: ‌
8!
4!4!0!
⋅24 Step 5: Calculate Coefficients Total Coefficient ‌=‌
8!
6!2!
⋅22+‌
8!
5!2!1!
⋅22⋅21+‌
8!
4!4!
⋅24 ‌=28×4+168×4×2+70×16 ‌=112+1344+1120 ‌=2576 The coefficient of x12 is 2576 .
