Question

find the zeroes of the quadratic polynomial x² – 2x – 8 and verify a relationship between zeroes and its coefficients

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients

x² – 2x – 8

Answer 1

f(x) = 𝑥² − 2𝑥 − 8

𝑓(𝑥) = 𝑥² − 2𝑥 − 8

= 𝑥² − 4𝑥 + 2𝑥 − 8

= 𝑥(𝑥 − 4) + 2(𝑥 − 4)

= (𝑥 + 2)(𝑥 − 4)

Zeroes of the polynomials are -2 and 4

Sum of the zeroes `="-coefficient of x"/"coefficient of x"`

`-2+4=(-(-2))/1`

2 = 2

Product of the zeroes `="constant term"/("coefficient of "x^2)`

`-2xx4=(-8)/1`

`-8 = -8`

Answer 2

x² - 2x- 8 = x² - 4x + 2x - 8

= x(x-4) + 2(x - 4)

= (x - 4)(x + 2)

Therefore the zeroes of the polynomial x² - 2x - 8 are {4, 2}

Relationship between the zeroes and the coefficients of the polynomial

Sum of the zeroes = - `(`

Also sum of the zeroes of the polynomial = 4 - 2 = 2

Product of the zeroes = `(`

Also product of the zeroes  = 4 x -2 = -8

Hence verified