Weightage: 3 – 4 marks
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Concepts covered:
- Zeroes of a polynomial
- Geometrical meaning of zeroes of a polynomial
- Relation between the zeroes and coefficients of a polynomial
- Division Algorithm for Polynomials
Expected questions:
- Questions based on finding zeroes of or using zeroes to find an unknown value are asked for 1 or 2 marks.
- Questions based on relation between the zeroes and coefficients of a polynomial or division algorithm for polynomials are asked for 3 or 4 marks.
Synopsis:
- Zeroes of a polynomial: A real number k is said to be a zero of the polynomial p(x) if p(k)=0, where, p(k) is obtained by substituting k for x in the polynomial.
- Geometrical meaning of zeroes of a polynomial: 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.
- Relation between the zeroes and coefficients of a polynomial:
- If α and β are the zeroes of a quadratic polynomial, ax2+bx+c then α+β= -b/a and αβ= c/a
- If α, β and γ are the zeroes of a cubic polynomial, ax3+bx2+cx+d then α+β+γ= -b/a, αβ+βγ+γα= c/a and αβγ= -d/a
- Division algorithm for polynomials: 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).