Question

on dividing the polynomial x³+2x²+kx+7 by x-3, remainder comes out to be 25. Find quotient and the value of k. Also find the sum and product of zeroes of the quotient so obtained.​

Answer 1

Step-by-step explanation:

Given :-

On dividing the polynomial x³+2x²+kx+7 by x-3, remainder comes out to be 25.

To find :-

1)Find quotient

2)Find the value of k.

3) Find the sum and product of zeroes of the quotient so obtained.

Solution :-

Given Polynomial is x³+2x²+kx+7

Given divisor = x-3

Given remainder = 25

We know that

Remainder Theorem

If P(x) is divided by x-a then the remainder is P(a)

we have

p(x) = x³+2x²+kx+7

If P(x) is divided by (x-3) then the remainder is P(3)

=> P(3) = 3³+2(3)²+k(3)+7

=> P(3) = 27+2(9)+3k+7

=> P(3) = 27+18+3k+7

=> P(3) = 52+3k

According to the given problem

The remainder =25

=> P(3) = 25

=> 52+3k = 25

=> 3k = 25-52

=> 3k = -27

=> k = -27/3

=> k = -9

therefore,k = -9

If k = -9 then the given Polynomial will be

x³+2x²-9x+7

We know that Division Algorithm on Polynomials

p(x) = g(x)q(x)+r(x)

p(x) = x³+2x²-9x+7

g(x) = x-3

q(x) = quotient

r(x) = 25

On Substituting these values in the above formula

=> x³+2x²-9x+7 = (x-3)×q(x)+25

=> x³+2x²-9x+7-25 = (x-3)×q(x)

=> x³+2x²-9x-18 = (x-3)×q(x)

=> (x-3)×q(x) = x³+2x²-9x-18

=> q(x) = (x³+2x²-9x-18)/(x-3)

x-3 ) x³+2x²-9x-18 (x²+5x+6

x³-3x²

(-) (+)

____________

5x²-9x

5x²-15x

(-) (+)

______________

6x -18

6x-18

(-) (+)

_______________

0

_______________

Quotient = x²+5x+6

On Comparing this with the standard quadratic polynomial ax²+bx+c

a = 1

b = 5

c = 6

We know that

sum of the zeroes = -b/a

=> -5/1

=> -5

Sum of the zeroes = -5

Product of the zeroes = c/a

=> 6/1

=> 6

product of the zeroes = 6

Answer:-

1) Value of k for the given problem is -9

2) The quotient for the given problem is x²+5x+6

3) The sum of the zeroes = -5

4) The Product of the zeroes = 6

Used formulae:-

Remainder Theorem:-

• Let P(x) be a polynomial of the degree greater than or equal to 1 and x-a is another linear polynomial if P (x) is divided by x-a then the remainder is P(a).

Division Algorithm on Polynomials:-

• p(x) = g(x)q(x)+r(x)

Where,

• p(x) = Dividend

• g(x) = Divisor

• q(x) = Quotient

• r(x) = Remainder

• The standard quadratic polynomial is ax²+bx+c

• Sum of the zeroes = -b/a

• Product of the zeroes = c/a