Relationship between Zeroes and Coefficients of a Polynomial
You have already seen that zero of a linear polynomial ax + b is
So, we write
So, the value of p(x) =
i.e., when x = 1 or x = 3. So, the zeroes of
Sum of its zeroes = 1 + 3 =
Product of its zeroes = 1 × 3 =
In general, α and β are the zeroes of the quadratic polynomial p(x) =
a
= k[
= k
Comparing the coefficients of
a = k, b = – k(α + β ) and c = kαβ
This gives α + β =
αβ =
Let us take one more quadratic polynomial, say, p(x) =
By the method of splitting the middle term,
Since p(x) can have atmost three zeroes, these are the zeores of
4,-2 and
Sum of zeros = 4 + (-2) +
Product of zeros =
For cubic polynomials we can also get one more identity. Mix the zeros two at a time and we get the following:
In general, it can be proved that if α, β, γ are the zeroes of the cubic polynomial
2. Find the zeroes of the quadratic polynomial
3. Find the zeroes of the polynomial
4. Verify that 3, –1,
- Comparing the given polynomial with
+ax 3 + cx + d. we get a =bx 2 , b = , c = , d = - Further: p(3) = 3 ×
- (5 ×3 3 ) - (11 × 3) -3 =3 2 – – – (substitute the value of '3' in given eq) - Hence we get the answer is
- p(-1)= 3 ×
– 5 ×− 1 3 – 11 × (–1) – 3 = –− 1 2 – + – (substitute the value of '-1' in given eq) - Hence we get the answer is
- p(
− ) = 3 ×1 3 - 5 ×− 1 3 3 - 11 ×− 1 3 2 - 3 =− 1 3 + + + (substitute the value of − in given eq)1 3 - Hence we get the answer is
− +2 3 =2 3 - Therefore, 3, –1 and
− are the zeroes of1 3 3 -x 3 5 – 11x – 3x 2 - So, we take α = 3, β= -1, γ =
− 1 3 - Now, α + β + γ = 3 + (-1) + (
− ) = 2 -1 3 =1 3 - αβ + βγ + γα = 3 × (-1) + (-1) × (
− ) + (1 3 − ) × 3 = 3 +1 3 - 1 =1 3 - αβγ = 3 × (-1) × (
− ) =1 3 - We found the required answers.