The number of zeroes of a polynomial is the number of points where the graph of $y = p(x)$ intersects the $x$-axis. Counting the intersections in Fig. 2.10 gives: (i) no intersection, (ii) one intersection, (iii) three intersections, (iv) two intersections, (v) four intersections, and (vi) three intersections.
(i) 0
(ii) 1
(iii) 3
(iv) 2
(v) 4
(vi) 3
(i) $x^2 - 2x - 8 = (x - 4)(x + 2)$, so the zeroes are $4$ and $-2$.
(ii) $4s^2 - 4s + 1 = (2s - 1)^2$, so the zeroes are $\dfrac{1}{2}$ and $\dfrac{1}{2}$.
(iii) $6x^2 - 3 - 7x = 6x^2 - 7x - 3 = (3x + 1)(2x - 3)$, so the zeroes are $-\dfrac{1}{3}$ and $\dfrac{3}{2}$.
(iv) $4u^2 + 8u = 4u(u + 2)$, so the zeroes are $0$ and $-2$.
(v) $t^2 - 15 = (t - \sqrt{15})(t + \sqrt{15})$, so the zeroes are $\sqrt{15}$ and $-\sqrt{15}$.
(vi) $3x^2 - x - 4 = (3x - 4)(x + 1)$, so the zeroes are $\dfrac{4}{3}$ and $-1$.
For a quadratic polynomial $az^2 + bz + c$, the relationships are sum of zeroes $= -\dfrac{b}{a}$ and product of zeroes $= \dfrac{c}{a}$; the values above verify these in each case.
(i) Zeroes: $4, -2$; sum $= 2 = -\dfrac{-2}{1}$, product $= -8 = \dfrac{-8}{1}$.
(ii) Zeroes: $\dfrac{1}{2}, \dfrac{1}{2}$; sum $= 1 = -\dfrac{-4}{4}$, product $= \dfrac{1}{4} = \dfrac{1}{4}$.
(iii) Zeroes: $\dfrac{3}{2}, -\dfrac{1}{3}$; sum $= \dfrac{7}{6} = -\dfrac{-7}{6}$, product $= -\dfrac{1}{2} = \dfrac{-3}{6}$.
(iv) Zeroes: $0, -2$; sum $= -2 = -\dfrac{8}{4}$, product $= 0 = \dfrac{0}{4}$.
(v) Zeroes: $\sqrt{15}, -\sqrt{15}$; sum $= 0 = -\dfrac{0}{1}$, product $= -15 = \dfrac{-15}{1}$.
(vi) Zeroes: $\dfrac{4}{3}, -1$; sum $= \dfrac{1}{3} = -\dfrac{-1}{3}$, product $= -\dfrac{4}{3} = \dfrac{-4}{3}$.
If the sum of zeroes is $S$ and the product is $P$, one required quadratic polynomial is $x^2 - Sx + P$; any non-zero constant multiple is also valid.
(i) $S = \dfrac{1}{4}$, $P = -1$: $x^2 - \dfrac{1}{4}x - 1$, or $4x^2 - x - 4$.
(ii) $S = \sqrt{2}$, $P = \dfrac{1}{3}$: $x^2 - \sqrt{2}x + \dfrac{1}{3}$, or $3x^2 - 3\sqrt{2}x + 1$.
(iii) $S = 0$, $P = \sqrt{5}$: $x^2 + \sqrt{5}$.
(iv) $S = 1$, $P = 1$: $x^2 - x + 1$.
(v) $S = -\dfrac{1}{4}$, $P = \dfrac{1}{4}$: $x^2 + \dfrac{1}{4}x + \dfrac{1}{4}$, or $4x^2 + x + 1$.
(vi) $S = 4$, $P = 1$: $x^2 - 4x + 1$.
(i) $4x^2 - x - 4$
(ii) $3x^2 - 3\sqrt{2}x + 1$
(iii) $x^2 + \sqrt{5}$
(iv) $x^2 - x + 1$
(v) $4x^2 + x + 1$
(vi) $x^2 - 4x + 1$