Question

$determine \: which \: of \: the \: following \: polynomials \: has \: x - 2 \: a \: factor.$
3x^2+6x-24​

Answer 1

Answer :

(x - 2) is a factor of the given polynomial 3x² + 6x - 24.

Step-by-step explanation :

➤ Quadratic Polynomials :

✯ It is a polynomial of degree 2

✯ General form :

          ax² + bx + c  = 0

✯ Determinant, D = b² - 4ac

✯ Based on the value of Determinant, we can define the nature of roots.

        D > 0 ; real and unequal roots

        D = 0 ; real and equal roots

        D < 0 ; no real roots i.e., imaginary

✯ Relationship between zeroes and coefficients :

          ✩ Sum of zeroes = -b/a

          ✩ Product of zeroes = c/a

________________________________

Given polynomial,

3x² + 6x - 24

METHOD - 1 :

Factorizing,

   = 3x² + 6x - 24

   = 3 (x² + 2x - 8)

   = 3 (x² - 2x + 4x - 8)

   = 3 (x(x - 2) + 4(x - 2))

   = 3 (x - 2) (x + 4)

∴ (x - 2) is a factor of the given polynomial.

METHOD - 2 :

• (x - 2) is a factor

   => x - 2 = 0

          x = 2

If it is a factor, when we substitute x = 2, the result is zero.

⇒ 3x² + 6x - 24

⇒ 3(2)² + 6(2) - 24

⇒ 3(4) + 12 - 24

⇒ 12 + 12 - 24

⇒ 24 - 24

⇒ 0

The result is zero.

∴ (x - 2) is a factor of the given polynomial.