Question

the cubic polynomial f(x) is such that the coefficient of x^3 is -1 and the zeroes of f(x) are 1, 2 and k. if f(x) has th remainder 8 when divided by x-3, then find the value of k

Answer 1

Answer:

1)k = 7

2)200

Step-by-step explanation:

Given the zeros of f(x) are 1,2 and k.

Any cubic polynomial with 3 roots p,q and r  and leading coefficient a will be in the form

a(x-p)(x-q)(x-r) = 0

Given f(x) has leading coefficient -1 so a = -1

Given the zeros p =1 , q =2 and r = k

Hence f(x) will be of the form

f(x) = -(x-1)(x-2)(x-k)

Now, given that f(x) has remainder of 8 when divided by x-3

We know from Remainder's Theorem , when f(x) is divided by (x-a), then the remainder will be equal to f(a).

Hence f(3) = 8

But f(3) = -(2)(1)(3-k) = 8

=>3-k = -4

=>k = 7-----Ans

2)

So, f(x) = -(x-1)(x-2)(x-7)

Now, if f(x) is divided by (x+3), remainder will be equal to

=f(-3)

=-(-4)(-5)(-10)

=200.-----Ans

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Answer 2

Answer:

Given the zeros f(x) are 1,2 and k.

Any cubic polynomial with 3 roots p, q and r and leading coefficient a will be in the form

a(x-p) (x-q) (x-r) = 0

Given f(x) has leading coefficient -1 so a = -1

Given the zeros p =1, q =2 and r = k

Hence f(x) will be of the form

f(x) = -(x-1) (x-2) (x-k)

Now, given that f(x) has remainder of 8 when divided by x-3

We know from Remainder's Theorem, when f(x) is divided by (x-a), then the remainder will be equal to f(a).

Hence f (3) = 8

But f (3) = -(2)(1) (3-k) = 8

=>3-k = -4

=>k = 7-----Ans

2)

So, f(x) = -(x-1) (x-2) (x-7)

Now, if f(x) is divided by (x+3), remainder will be equal to

=f (-3)

=-(-4) (-5) (-10)

=200. -----Ans

Step-by-step explanation: