Question

Find the zeroes of the polynomial and verify the relations between coefficient and zeroes

3x²-x-4

Answer 1

(-1) & 4/3

Explanation:

Polynomial:

3x² - x - 4

Sum = -1

Product = 3*(-4) = -12

Factors = -4 and 3 { (-4)*3 = -12 & (-4) + 3 = -1}

3x² - x - 4 = 3x² + 3x - 4x -4 {Rewrite the middle term using the factors}

= 3x(x + 1) -4(x + 1)

=(x + 1)(3x - 4)

x + 1 = 0 ; 3x - 4 = 0

x = -1 ; 3x = 4

x = 4/3

`\sf Zeroes \ of \ the \ polynomial \ are \ (-1) \ and \ (4)/(3)`

Verification:

`\sf 3x^2 - x - 4 = 0\\\\Divide \ the \ entire \ equation \ by \ 3\\\\\\(3x^2)/(3)-(x)/(3)-(4)/(3)=0\\x^2 - (x)/(3)-(4)/(3)=0`

Coefficient of x = sum of the zeroes

`\sf = -1 + (4)/(3)\\\\=(-3)/(3)+(4)/(3)\\\\=(1)/(3)\\\\`

Constant = product of the zeroes

`\sf = (-1)*(4)/(3)\\\\=(-4)/(3)`

Thus verified.