Example 2
Find the zeroes of the quadratic polynomial x2 + 7x + 10, and verify the relationship between the zeroes and the coefficients.
Let p(x) = x2 + 7x + 10
Zero of the polynomial is the value of x where p(x) = 0
Putting p(x) = 0
x2 + 7x + 10 = 0
We find roots using splitting
the middle term method
x2 + 2x + 5x + 10 = 0
x(x + 2) + 5(x + 2) = 0
(x + 2)(x + 5) = 0
So, x = −2, −5
Therefore,
α = -2 & β = −5 are the zeroes of polynomial
Splitting the middle term method
We need to find two numbers whose
Sum = 7
Product = 10 × 1 = 10
Verifying relationship b/w zeroes and coefficients
p(x) = x2 + 7x + 10 = 1x2 + 7x + 10
Comparing with ax2 + bx + c
a = 1 ,
We have to verify
Sum of zeroes = − (𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑥)/(𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑥2)
i.e. α + β = - 𝒃/𝒂
Product of zeroes = (𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑡𝑒𝑟𝑚)/(𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑥2)
i.e. α × β = 𝒄/𝒂
α + β
= −2 + (−5)
= −2 − 5
= –7
− 𝑏/𝑎
= − 7/1
= −7
α β
= (−2) ( −5)
= 10
𝑐/𝑎
= 10/1
= 10
Since, L.H.S = R.H.S
Hence relationship between zeroes & coefficient is verified

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo

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