(i) Graph of a linear polynomial ax + b is a straight line.

(ii) Graph of a quadratic polynomial p(x) = ax^{2} + bx + c is a parabola open upwards like ∪, if a > 0.

(iii) Graph of a quadratic polynomial p(x) = ax^{2} + bx + c is a parabola open downwards like ∩, if a > 0.

(iv) In general a polynomial p(x) of degree n crosses the x-axis at atmost n points.

(i) If α, β are zeroes / roots of p(x) = ax2 + bx + c, then

(ii) If α, β and γ are zeroes / roots of p(x) = ax^{3} + bx^{2} + cx + d

(iii) If α, β are roots of a quadratic polynomial p(x), then p(x) = x2 – (α + β) x + αβ

⇒ p(x) = x^{2} – (sum of roots) x + product of roots

(iv) If α, β and γ are zeroes of a cubic polynomial p(x),

Then, p(x) = x3 – (α + β + γ) x^{2} + (αβ + βγ + αγ) x – (αβγ)

⇒ p(x) = x^{3} – (sum of zeroes) x^{2} + (sum of product of zeroes / roots taken two at a time)

x – (product of zeroes)

⇒ p(x) = (x + 3)(x + 4)

∴ p(x) = 0 if x + 3 = 0 or x + 4 = 0

⇒ x = – 3 or x = – 4

∴– 3 and – 4 are zeros of the p(x).

Here coefficient of x^{2} = 4,

Coefficient of x = 0 and constant term = –7.

⇒α + β = Sum of zeroes

⇒α + β = 0 ⇒ 5 + β = 0 ⇒β = –5

Now product of zeroes = αβ = 5 × (–5) = –25

Let polynomial p(x) = ax^{2} + bx + c

and &alpha, &beta, γ are its zeroes.

∴ α + β + γ = Sum of zeroes

αβ + αγ + βγ = Sum of the products of zeroes taken two at a time

If a = 1, then b = –5, c = –6 and d = 20

∴ Polynomial, p(x) = x^{3} – 5x^{2} – 6x + 20.

If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that

p(x) = q(x) × g(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x).

∴ Quotient is 2x – 2 and remainder is 9x – 4.

4x^{4} – 20x^{3} + 23x^{2} + 5x – 6 if two of its zeroes are 2 and 3.

∴ (x – 2)(x – 3) is a factor of given polynomial.

⇒ x^{2} – 5x + 6 is a factor of given polynomial.

Now

∴ 4x^{4} – 20x^{3} + 23x^{2} + 5x – 6

= (x^{2} – 5x + 6)(4x^{2} – 1)

= (x – 2)(x – 3)(2x – 1)(2x + 1)

∴ Zeroes of the given polynomial are