Question

Find the zeroes of the polynomial x^3-6x^2+3x+10, it is given that the zeroes are in AP

Answer 1

Given :

• x³ - 6x² + 3x + 10
• Zeroes are in AP

To Find :

• Zeroes of the polynomial.

Solution :

Take four numbers -1,1,-2,2.

Choose a number, for example 1.

Put the value of number chosen in the polynomial in place of variables. Solve the polynomial after placing the number in place of variables and check if you get zero, after performing mathematical operation.

In the above polynomial, the variable is x and number chosen from the four number mentioned is 1.

$\sf{\longrightarrow{x^3-6x^2+3x+10}}$

$\sf{\longrightarrow{(1)^3-6(1)^2+3(1)+10}}$

$\sf{\longrightarrow{1-6+3+10}}$

$\sf{\longrightarrow{-5+13}}$

$\sf{\longrightarrow {8}}$

Here after adding / subtracting the end result isn't zero.

•°• 1 is not a factor.

Now, let's try with -1.

$\sf{\longrightarrow{x^3-6x^2+3x+10}}$

$\sf{\longrightarrow{(-1)^3-6(-1)^2+3(-1)+10}}$

$\sf{\longrightarrow{-1-6-3+10}}$

$\sf{\longrightarrow{-7+7}}$

$\sf{\longrightarrow{0}}$

•°• (x+1) is one of the three zeroes of the polynomial.

Now the next step is to write down the coefficients of the polynomial and use synthetic division.

[Refer attachment]

Copy the first coefficient as it is, in this case 1. Then multiply diagonally with the factor and keep adding the product.

-1 × 1 = - 1

[We add it to -6 and ingrained -7]

Next, -7 × -1 = 7

[ Adding 7 to the next coefficient i.e 3, we get 10]

Next, -1 × 10 = - 10

[ Adding to the last coefficient i.e 10, we get zero]

Now, the numbers obtained after adding the product of multiplication, 1, -7 and 10 are actually the coefficient of the quadratic polynomial.

It forms a quadratic polynomial,

$\bold{x^2-7x+10}$

Using factorization method we can find zeroes of the polynomial.

$\sf{\longrightarrow{x^2-7x+10=0}}$

$\sf{\longrightarrow{x^2-5x-2x+10=0}}$

$\sf{\longrightarrow{x(x-5)-2(x-5) =0}}$

$\sf{\longrightarrow{(x-5) =0\:\:\:or\:\:\:(x-2) =0}}$

$\sf{\longrightarrow{x=5\:\:\:or\:\:\:x=2}}$

We have the three zeroes of the cubic polynomial,

$\large{\boxed{\sf{x=-1,2,5}}}$

The common difference between the first zero, second zero and third zero is constant. Common difference is 3.

$\sf{Second\:zero\:-\:First\:zero\:=\:2-(-1)=3}$

$\sf{Third\:zero-\:second\:zero\:=\:5-2=3}$

•°• The zeroes are in AP.