**Question:**1. If 5+2√3

——— = a+b √3, find the value of "a" and "b"

7+4√3

2. If f(x)=x▲4 - 2x▲3 + 3x▲2 - ax + b is a polynomial such that when it is divided by x -1 and x +1, the remainders are respectively 5 and 19. Determine the values of "a" and "b"

NOTE: x▲4 means "x to the power of 4" i.e, 4 is the exponent of x. Same goes to the other numbers.

**Answers:**

1) Multiply the numerator and the denominator by the conjugate of (7+4√3), i.e. by (7-4√3). Then, by the formula (a+b)(a-b)=a^2-b^2 you will get:

(5+2√3) (7-4√3)

——————— = 35-20√3+14√3-24 = 11 - 6√3

49 - 16(3)

So, a=11, b=6

2) f(1)=5, f(-1)=19 =>

1-2+3-a+b=5 <=> -a+b=3

and

1+2+3+a+b=19 <=> a+b=19

Solving these two equations, you obtain:

b=11, a=8

what's √?

1) mutiply and divide the eq by 7- 4√3 , u get 11 - 6√3 , so a = 11 and b = -6.

2) f(1) and f(-1) should be equal to zero, so substitute x = 1 and -1 , u get 2 eq in 2 variables solve 'em.

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