Question

two zeros of a cubic polynomial ax^3+ 3x^2-bx-6 are -1 and -2. Find the third zero and the value of a and b

Answer 1

$\Large{\underline{\underline{\mathfrak{\bf{Question}}}}}$

two zeros of a cubic polynomial ax^3+ 3x^2-bx-6 are -1 and -2. Find the third zero and the value of a and b

$\Large{\underline{\underline{\mathfrak{\bf{Solution}}}}}$

$\Large{\underline{\mathfrak{\bf{\pink{Given}}}}}$

• Polynomial , ax³ + 3x² - bx - 6 = 0 .........(1)
• -1 and -2 are zeroes

$\Large{\underline{\mathfrak{\bf{\pink{Find}}}}}$

• Value of a, b and third zeroes

$\Large{\underline{\underline{\mathfrak{\bf{Explanation}}}}}$

Here, -1 and -2 are zeroes of this equation (1),

So, x = -1 , -2 exits of this equation (1),

If, any value of x does not exits of this equation,

so, we can say that this value of x is not zeroes of this equation .

Case(1):-

• X = -1, keep in equ(1)

➠ a(-1)³+3.(-1)²-b.(-1)-6 = 0

➠ -a + 3 + b -6 = 0

➠ a - b = -3 .............(2)

Case(2):-

• X = -2, keep in equ(2)

➠ a(-2)³+3.(-2)²-b.(-2)-6 = 0

➠ -8a + 12 + 2b - 6 = 0

➠ 8a - 2b = 6

➠4a - b = 3 ...............(3)

Sub. equ(2) and equ(3)

➠ (a - 4a ) = (-3-3)

➠ -3a = -6

➠ a = -6/(-3)

➠ a = 2

Keep value of a in equ(2),

➠ 2 - b = -3

➠ b = 2 + 3

➠ b = 5

Thus:-

• Value of a = 2
• Value of b = 5

$\Large{\underline{\underline{\mathfrak{\bf{Answer\:verification}}}}}$

Keep value of a and b in equ(2),

➠ 2 - 5 = -3

➠ -3 = -3

L.H.S. = R.H.S.

Hence, we can say that value of a and b are absolutely right

___________________________

Now, keep value of a and b in equ(1),

➠ (2).x³ + 3x² - (5).x - 6 = 0

➠ 2x³ + 3x² - 5x - 6 = 0

➠2x³ + 2x² + x² + x - 6x -6 = 0

➠ 2x²(x+1)+x(x+1)-6(x+1) = 0

➠ (x+1)(2x²+x-6) = 0

➠ (x+1)[2x² - 3x + 4x - 6 ] = 0

➠ (x+1)[2x(x + 2) -3(x+ 2) ] = 0

➠ (x+1)(x+2)(2x-3) = 0

So, Third Zeroes will be

➠ (2x - 3 ) = 0

➠ 2x = 3

➠ x = 3/2

Hence, required Third zeroes is 3/2 .