Question

Without actual division , prove that (2X4-6X3+3X2+3X-2) is exactly divisible by (X2-3X+2).

Answer 1

Hii friend,


Let P(X) = 2X⁴-6X³+3X²+3X-2

Let G(X) = (X²-3X+2) = (X²-2X-X+2)

=> X(X-2) -1(X-2)

=> (X-2) (X-1).

Now, P(X) will be exactly divisible by G(X) if it is exactly divisible by (X-2) as well as (X-1).

Putting X = 2 in P(X).

P(X) = 2X⁴-6X³+3X²+3X-2

P(2) = ( 2 × 2⁴ - 6 × 2³ + 3 × 2² + 3 × 2 -2)

=> (32-48+12+6-2) = (50-50) = 0

And,

P(1) = (2 × 1⁴ -6 × 1³ + 3 × 1² + 3 × 1 -2)

=> (2-6+3+3-2) = (8-8) = 0

Therefore,

P(X) is exactly divisible by (X-2) and (X-1)

So , P(X) is exactly divisible by (X²-3X+2)

Hence,

P(X) is exactly divisible by (X²-3X+2)

HOPE IT WILL HELP YOU..... :-)

Answer 2

Given,

A polynomial f(x) = 2x^4 - 6x^3 + 3x^2 + 3x - 2

To prove,

f(x) is divisible by p(x) = x^2 - 3x + 2, without actual division.

Solution,

We can simply prove this mathematical condition using the following process:

Mathematically,

If A and B are two factors of a number/polynomial, then the product of A and B, that is, AB is also a factor of the number/polynomial. Conversely, the number/polynomial is divisible by A, B, and AB. {Equation-1}

Now,

p(x) = x^2 - 3x + 2

= x^2 - 2x - x + 2

= x(x-2) - 1(x-2)

= (x-1)(x-2)

=> p(x) = x^2 - 3x + 2 = (x-1)(x-2) {Equation-2}

The given polynomial is;

f(x) = 2x^4 - 6x^3 + 3x^2 + 3x - 2

Now,

f(1) = 2(1)^4 - 6(1)^3 + 3(1)^2 + 3(1) - 2

= 2 - 6 + 3 + 3 - 2 = 0

=> f(1) = 0

=> (x-1) is a root of the polynomial f(x)

=> (x-1) is a factor of the polynomial f(x)

=> the polynomial f(x) is divisible by (x-1) {Equation-3}

f(2) = 2(2)^4 - 6(2)^3 + 3(2)^2 + 3(2) - 2

= 2(16) - 6(8) + 3(4) + 3(2) - 2

= 32 - 48 + 12 + 6 - 2 = 0

=> f(2) = 0

=> (x-2) is a root of the polynomial f(x)

=> (x-2) is a factor of the polynomial f(x)

=> the polynomial f(x) is divisible by (x-2) {Equation-4}

Now, combining the equations 1, 3, and 4, we conclude;

the polynomial f(x) is divisible by (x-1)(x-2)

=> the polynomial f(x) is divisible by x^2 - 3x + 2

( from equation-2)

=> the polynomial (2x^4 - 6x^3 + 3x^2 + 3x - 2) is divisible by (x^2 - 3x + 2)

=> the polynomial f(x) is divisible by p(x)

Hence, it is proved that f(x) is divisible by p(x) = x^2 - 3x + 2, without actual division.