Relationship between Zeroes and Coefficients of a Polynomial

You have already seen that zero of a linear polynomial ax + b is −ba . We will now try to answer the question raised in the previous section regarding the relationship between zeroes and coefficients of a quadratic polynomial. For this, let us take a quadratic polynomial, say p(x) = 2x2-8x + 6. Previously, you have learnt how to factorise quadratic polynomials by splitting the middle term. So, here we need to split the middle term ‘– 8x’ as a sum of two terms, whose product is 6 × 2x2 = x2

So, we write 2x2 -8x + 6 = 2x2 -6x -2x + 6 = 2x(x – 3) – 2(x – 3)= (2x – 2)(x – 3) = 2(x – 1)(x – 3)

So, the value of p(x) = 2x2 -8x + 6 = is zero when x – 1 = 01 or x – 3 = 01

i.e., when x = 1 or x = 3. So, the zeroes of 2x2 -8x + 6 are 1 and 3. Observe that :

Sum of its zeroes = 1 + 3 = = -(-8)2 = −Coefficient of xCoefficient ofx2

Product of its zeroes = 1 × 3 = = 62 = Constant termCoefficient ofx2

In general, α and β are the zeroes of the quadratic polynomial p(x) = ax2 + bx +c a ≠ 0, then you know that x – α and x – β are the factors of p(x). Therefore,

ax2 + b x + c = k(x – α ) (x – β), where k is a constant

= k[x2– ( α+ β)x + αβ]

= kx2 - k( α+ β) x + kαβ

Comparing the coefficients of x2, x and constant terms on both the sides, we get

a = k1, b = −kα+βkα+β and c = kαβαβ

This gives α + β = −baca (since a = k)

αβ = ca−ba

Let us take one more quadratic polynomial, say, p(x) = 3x2+5x−2.

By the method of splitting the middle term,

Since p(x) can have atmost three zeroes, these are the zeores of 2x3−5x2−14x+8.

4,-2 and 12 are the zeroes of that equation

Sum of zeros = 4 + (-2) + 12 = 52 = −−52 52 = −baba.

Product of zeros = 4×−2×12= -4 = −8282 = −dada.

For cubic polynomials we can also get one more identity. Mix the zeros two at a time and we get the following:

4×−2+4×12+−2×12= -8 + 2 - 1 = -7 = -142142= ca−ca.

2. Find the zeroes of the quadratic polynomial x2 + 7x + 10, and verify the relationship between the zeroes and the coefficients.

Instruction

3. Find the zeroes of the polynomial x2 -3 and verify the relationship between the zeroes and the coefficients.

Instruction