Six graphs of y = p(x) are given in the textbook. In each case, state how many zeroes the polynomial p(x) has.
Hint. A zero is an x-value where p(x) = 0 — that is, a point where the curve actually touches or crosses the x-axis. Count those points.
There is one idea behind all six parts, so get it straight first and the counting becomes trivial.
The principle. A zero of p(x) is a value of x for which p(x) = 0. On the graph, p(x) = 0 means the height is zero — the curve is sitting on the x-axis. So:
the number of zeroes = the number of points where the graph meets the x-axis.
It does not matter whether the curve crosses through the axis or just touches it and turns back — both count as meeting it. What does not count is meeting the y-axis, which trips up a lot of students.
Applying it to the six graphs.
(i) The curve stays entirely on one side of the x-axis and never reaches it. No meeting points, so 0 zeroes.
(ii) The curve crosses the x-axis at exactly one point. 1 zero.
(iii) The curve cuts the x-axis at three separate points. 3 zeroes.
(iv) The curve meets the x-axis at two points. 2 zeroes.
(v) The curve meets the x-axis at four points. 4 zeroes.
(vi) The curve meets the x-axis at three points. 3 zeroes.
✦ Answer: (i) 0, (ii) 1, (iii) 3, (iv) 2, (v) 4, (vi) 3.
Where students slip. Counting where the graph crosses the *y*-axis. Every function crosses the y-axis at most once and it tells you the constant term, not a zero. Zeroes live on the x-axis only.
Another way. Useful cross-check: a polynomial of degree n has at most n zeroes. So a graph meeting the axis four times cannot come from a quadratic — it must be degree 4 or higher.
