Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes

(i) 2x³ + x² – 5x + 2; ¹/₂ , 1, -2

p(x) = 2x³ + x² – 5x + 2

Zeros 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(1)³ + (1)² – 5(1) + 2 = 0

p(-2) = 2(-2)³ + (-2)² – 5(-2) + 2

= -16 + 4 + 10 + 2

= 0

Therefore, ½ , 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

Let us take α = ¹/₂ ,β = 1, γ = -2

α + β + γ = ¹/₂ + 1 + (-2) = –¹/₂ = –^b/_a

αβ + βγ + γα = ¹/₂ x 1 + 1(-2) + ¹/₂ (-2) = ^-5/₂ = ^c/_a

αβγ = ¹/₂ x 1 x (-2) = –¹/₁ = – ²/₂ = –^d/_a

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

(ii) x³ – 4x² + 5x – 2; 2,1,1

p(x) = x³ – 4x² + 5x – 2

Zeros for this polynomial are 2 , 1, 1

p(2) = 2³ – 4(2)² + 5(2) – 2

= 8 – 16 + (10) – 2

= 0

p(1) = (1)³ – 4(1)² + 5(1) – 2 = 0

Therefore, 2 , 1, and 1 are the zeroes of the given polynomial. Comparing the given polynomial with ax³ + bx² + cx + d, we obtain a = 1, b = -4, c = 5, d = -2

Verify:

Sum of zeros = 2+1+1 = 4 = – ^(-4)/₁ = –^b/_a

Multiplication of zeroes taking two at a time = (2)(1)+(1)(1)+(2)(1) = 2+1+2= 5 = ⁵/₁ = ^c/_a

Multiplication of zeroses = 2 x 1 x 1 = 2 = – ^(-2)/₁ = –^d/_a

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