2. Polynomials

Nothing in the world is more common than unsuccessful people with talent. – Unknown

In this chapter, you have studied the following points:
1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.

2. A quadratic polynomial in x with real coefficients is of the form ax^2+ bx + c, \text{ where } a, b, c are real numbers with a ≠ 0.

3. The zeroes of a polynomial p(x) are precisely the x-coordinates of the points, where the graph of y = p(x) intersects the x -axis.

4. A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have at most 3 zeroes.

5. If α and β are the zeroes of the quadratic polynomial ax^2+bx+c=0 then \alpha + \beta = -\frac{b}{a}, \alpha \beta = \frac{c}{a} .

6. If \alpha, \beta \gamma are the zeroes of the cubic polynomial ax^3+bx^2+cx+d, then
\alpha+\beta \gamma = \frac{-b}{a},
\alpha \beta + \beta \gamma + \gamma \alpha = \frac{c}{a}
and \alpha \beta \gamma = \frac{-d}{a}.

7. The division algorithm states that given any polynomial p(x) and any non-zero polynomial g(x), there are polynomials q(x) and r(x) such that p(x) = g(x) q(x) + r(x),
where r(x) = 0 or degree r(x) < degree g(x).

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