Example 5 (Optional)
Verify that 3, –1, (−1)/3 are the zeroes of the cubic polynomial p(x) = 3x3 – 5x2 – 11x – 3, and then verify the relationship between the zeroes and the coefficients.
Let p(x) = 3x3 − 5x2 − 11x − 3
Verifying zeroes
At x = 𝟑
p(𝟑) = 3 (3)^3 − 5 (3)^2
− 11 (3) − 3
= 3 (27) − 5 (9)
− 33 − 3
= 81 − 45 − 33 − 3
= 81 − 81
= 0
Since p(3) = 0
∴ 𝟑 is a zero of p(x)
At x = −𝟏
p(−1) = 3 (−1)3 − 5 (−1)2 – 11 (−1) − 3
= 3 (−1) − 5 (1) +
11 − 3
= − 3 − 5 + 11 − 3
= − 11 + 11
= 0
Since p(−1) = 0
∴ −1 is a zero of p(x)
At x = – 𝟏/𝟑
p((−𝟏)/𝟑) = 3 ((−1)/3)^3 −
5((−1)/3)^2 − 11 ((−1)/3) − 3
= (−1)/9 − 5/9 + 11/3 − 3
= (−1 − 5 + 33 − 27)/9
= (−33+33)/9
= 0
Since p((−1)/3) = 0
∴ ((−𝟏)/𝟑) is a zero of p(x).
Verifying relationship between zeroes and coefficients
For a cubic Polynomial
p(x) = ax3 + bx2 + cx + d
With zeroes α, 𝛽 and γ
We have
𝛂 + 𝛽 + 𝛄 = (−𝒃)/𝒂
𝛂"𝛽" + 𝛽𝛄 + 𝛄𝛂 = 𝒄/𝒂
𝛂"𝛽" 𝛄= (−𝒅)/𝒂
For p(x) = 3x3 − 5x2 − 11x − 3
a = 3, b = − 5, c = − 11 and d = − 3
And zeroes are
𝜶 = 3, 𝜷 = − 1 and 𝜸 = (−1)/3
Now
𝜶+ 𝜷 + 𝜸
= 3 + (−1) +((−1)/3)
= 3 − 1 − 1/3
= (9 − 3 − 1)/3
= 5/3
= (−𝒃)/𝒂
𝜶𝜷+ 𝜷𝜸 + 𝜸𝜶
= (3) (−1) + (−1) ((−1)/3)
+ ((−1)/3)(3)
= −3 + 1/3 − 1
= (− 9 + 1 − 3)/3
= (−11)/3
= 𝒄/𝒂
𝜶𝜷𝜸
= 3 × (−1) ×((−1)/3)
= 3/3
= 1
= (−𝒅)/𝒂
Hence, the relationship is verified

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.