Question

Verify that the numbers given along side of the cubic polynomials are their zeroes. Also verify the relationship between the zeroes and the coefficients.

`2x^3 + x^2 – 5x + 2 ; 1/2, 1, – 2`

Answer 1

p(x) = 2x³ + x² - 5x + 2

Zeroes for this polynomial are 1/2, 1, -2

p(1/2) = `2(1/2)^3+(1/2)^2-5(1/2)+2`

= `1/4+1/4-5/2+2`

=0

p(1) = 2 x 1³ + 1² - 5 x 1 + 2

= 0

p(-2) = 2(-2)³ + (-2)² - 5(-2) + 2

= -16 + 4 + 10 + 2 = 0

Therefore, 1/2, 1, and −2 are the zeroes of the given polynomial.

Comparing the given polynomial with ax³ + bx² + cx + d we obtain a = 2, b = 1, c = −5, d = 2

We can take α = 1/2,  β = 1, y = -2

α + β + γ = `1/2+1+(-2) = -1/2 = (-b)/a`

αβ + βγ + αγ = `1/2xx1+1(-2)+1/2(-2)=(-5)/2 = c/a`

αβγ = `1/2xx1xx(-2) = (-1)/1=(-(2))/2=(-d)/a`

Therefore, the relationship between the zeroes and the coefficients is verified.