Question

find the zero of the poly x³ - 5x²- 2x + 24 if it is given that the product of its two zeros is 12​

Answer 1

Answer:

zero=2

Step-by-step explanation:

product of two zeroes =12

a*b=12

product of zeroes =-d/a

a*b*c=-24/1

12*c=24

c=2

Answer 2

Given:

• There is a cubic polynomial x³-5x²-2x+24.
• Product of two of its zeroes is 12.

To Find:

• The zeroes of the polynomial.

Answer:

Given cubic polynomial is x³-5x²-2x+24.

We know that the product of zeroes of a cubic polynomial in standard form ax³+bx²+cx+d is given by :

• Product of roots = -d/a.

Also we must know that

• Sum of roots = -b/a.
• Sum of products of roots taken at once = c/a

Here if α, β and γ are the roots , then

• α β γ = -d/a.
• α + β + γ = -b/a.
• α β + β γ + α γ = c/a.

Let the product of two roots β γ be 12.

Atq ,

⇒ α β γ = -24/1.

⇒ 12 × α = -24

⇒ α = (-24)/12

⇒ α = (-2)

If (-2) is a zero of the polynomial then (x+2) is a factor of the polynomial. Now let's divide polynomial by (x+2).

[For division refer to attachment : ]

Now , x³-5x²-2x+24=(x+2)(x²-7x+12).

Let's factorise the quadratic polynomial now :—

= x² - 7x + 12.

= x² - 4x - 3x +12.

= x ( x - 4 ) -3 ( x -4).

= ( x - 3 ) ( x - 4).

Equate this with 0 ;

⇒ (x-3)(x-4) = 0.

⇒ x = 3,4

Hence the other two zeroes are 3 and 4.

Hence all zeroes of polynomial are (-2),3 & 4.