Q1.Answer the following and justify:
(i) Can x2 – 1 be the quotient on division of x6 + 2x3 + x – 1 by a polynomial
in x of degree 5?.
Solution:
(ii) What will the quotient and remainder be on division of ax2
+ bx + c by px3 + qx2 + rx + s, p ≠ 0?.
Solution:
(iii) If on division of a polynomial p (x) by a polynomial g (x), the quotient is zero, what is the relation between the degrees of p (x) and g (x)?.
Solution:
(iv) If on division of a non-zero polynomial p (x) by a polynomial g (x), the remainder is zero, what is the relation between the degrees of p (x) and g (x)?.
Solution:
(v) Can the quadratic polynomial x2 + kx + k have equal zeroes for some odd integer k > 1?.
Solution:
Q2. Are the following statements ‘True’ or ‘False’? Justify your answers.
(i) If the zeroes of a quadratic polynomial ax2 + bx + c are both positive, then a, b and c all have the same sign.
Solution:
Solution:
(ii) If the graph of a polynomial intersects the x-axis at only one point, it cannot be a quadratic polynomial.
Solution:
(iii) If the graph of a polynomial intersects the x-axis at exactly two points, it need not be a quadratic polynomial.
Solution:
(iv) If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.
Solution:
(v) If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign.
Solution:
(vi) If all three zeroes of a cubic polynomial x3 + ax2 – bx + c are positive, then at least one of a, b and c is non-negative.
Solution:
(vii) The only value of k for which the quadratic polynomial kx2 + x + k has equal zeros is 1/2.
Solution:
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