How do I find the value of a and b with the polynomial [math]f(x)=2x^3+ax^2-bx+3[/math] which has a factor [math]x+3[/math], and when divided by [math]x-2[/math], has a remainder of [math]15?[/math]
Let us see the polynomial f(x) = 2x^3 +ax^2 - bx + 3.(x+3) is one factor which means x = -3. Substitute -3 for x to get-54 +9a + 3b + 3 =0, or dividing by the HCF, which is 3 we get-18 + 3a + b +1 = 0, or3a + b = 17 … (1).When divided by (x-2) the polynomial yields a remainder of 15. So deduct 15 from the polynomial to get 2x^3 +ax^2 - bx + 3 - 15, or2x^3 +ax^2 - bx -12. This polynomial should be divisible by (x-2) without leaving a remainder. Hence x=2. Substitute 2 for x in2x^3 +ax^2 - bx -12, to get16 + 4a - 2b - 12 = 0, or4a - 2b = -4, or2a - b = -2 … (2).Add (1) and (2) to get5a = 15 or a = 3.Put that value of 3 for a in (1) to getb = 17 - 3a, or17 -9 = 8.Hence a = -3 and b = 8.Check: Re-write f(x) as 2x^3 +3x^2 - 8x + 3.If x = -3, f(3) = -54 +27 +24 + 3 = 0.If x = 2, f(2) = 16 +12 -16 + 3 = 15 which is the remainder as mentioned above.Hence a = -3 and b = 8.
√72 divided by √18 answer?
√72 / √18 √72/18) = √4 =2