Answer the following and justify:
(i) Can \(x^2-1\) be the quotient on division of \(x^6+2x^3+x-1\) by a polynomial in \(x\) of degree 5?
(ii) What will the quotient and remainder be on division of \(ax^2+bx+c\) by \(px^3+qx^2+rx+s\), \(p\ne0\)?
(iii) If on division of a polynomial \(p(x)\) by a polynomial \(g(x)\), the quotient is zero, what is the relation between \(\deg p\) and \(\deg g\)?
(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 \(\deg p\) and \(\deg g\)?
(v) Can the quadratic polynomial \(x^2+kx+k\) have equal zeroes for some odd integer \(k>1\)?
(i) No (ii) Quotient 0, remainder \(ax^2+bx+c\) (iii) \(\deg p<\deg g\) (iv) \(\deg p\ge \deg g\) (v) No
Are the following statements True/False? Justify:
(i) If the zeroes of a quadratic polynomial \(ax^2+bx+c\) are both positive, then \(a,b,c\) all have the same sign.
(ii) If the graph of a polynomial intersects the x-axis at only one point, it cannot be a quadratic polynomial.
(iii) If the graph of a polynomial intersects the x-axis at exactly two points, it need not be a quadratic polynomial.
(iv) If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.
(v) If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term have the same sign.
(vi) If all three zeroes of a cubic polynomial \(x^3+ax^2-bx+c\) are positive, then at least one of \(a,b,c\) is non-negative.
(vii) The only value of \(k\) for which the quadratic polynomial \(kx^2+x+k\) has equal zeroes is \(\dfrac{1}{2}\).
(i) False (ii) False (iii) True (iv) True (v) True (vi) False (vii) False