When the polynomial P(x)=ax^3+bx^2+3x-10 is divided by x+1, the remainder is -8. P(x) has a factor of x+5. Find the values of a and b.

a = 1, b = 6

Step-by-step explanation:

The equation given is as follows;

P(x) = a·x³ + b·x² + 3·x - 10

The above equation has a factor of x + 5, therefore, we have;

P(-5) = 0 = a·(-5)³ + b·(-5)² + 3·(-5) - 10

-125·a + 25·b + (-15) - 10 = 0

-125·a + 25·b - 25 = 0

-125·a + 25·b  = 25(1)

Also, we are given that;

a·x³ + b·x² + 3·x - 10 divided by x + 1 as a remainder, R = -8, therefore;

P(-1) = -8 = a·(-1)³ + b·(-1)² + 3·(-1) - 10

-a + b - 13 = -8

-a + b = -8 + 13 = 5

-a + b = 5(2)

Multiply equation (2) by 25 and subtract from (1) gives

-125·a + 25·b - 25(-a + b) = 25 - 25×5

-100·a = 25 - 125 = -100

a = 1

Therefore, from equation (2) we have;

-1 + b = 5

b = 5 + 1 = 6.

Step-by-step explanation:

Unfortunately it's not clear what you mean by 6 Σ 2 3(0.5)^n-1.  I am going to assume that you mean summing up from n = 2 to n = 6 of the expression 3(0.5)^(n-1).  Note that the "n-1" must be enclosed in parentheses.

a(2) = 3(0.5)^(2 -1) = 3(0.5) = 1.5

a(3) =  3(0.5)^(3 -1) = 3(0.5)^2 = 0.75

a(4) = 3(0.5)^(4 -1) = 3(0.5)^3 = 0.375

a(5) = 0.1875

a(6) = 0.09375

Sum these up to find the final answer.

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