NCERT Class 10 Mathematics — Polynomials, Exercise 2.1. The one question solved, all six parts.

This is a one-question exercise, and the question is about a single idea: the zeroes of a polynomial $p(x)$ are the values of $x$ for which $p(x) = 0$, and on the graph of $y = p(x)$ those are exactly the points where the curve meets the $x$-axis — because the $x$-axis is where $y = 0$.

So counting zeroes means counting meeting points. Two facts help you trust the count:

• A point where the curve only touches the $x$-axis and turns back still has $y = 0$ there, so it is a zero just as much as a crossing is.
• A polynomial of degree $n$ has at most $n$ zeroes — at most, not exactly. A cubic can have one, two or three.

Key insight. Look only at the $x$-axis. Where the curve crosses the $y$-axis, how high its humps are, and how many times it turns are all irrelevant. The answer is the number of points the curve shares with the $x$-axis — touching included — and nothing else.

Question 1

The graphs of $y = p(x)$ are given in Fig. 2.10 below, for some polynomials $p(x)$. Find the number of zeroes of $p(x)$, in each case.

The six graphs, redrawn from the textbook’s figure, with every point where the curve meets the $x$-axis marked:

Solution. Take each graph in turn and count the points it has in common with the $x$-axis.

(i) The graph is a straight line parallel to the $x$-axis, lying above it. It never meets the $x$-axis, so $p(x)$ is never $0$. (This is the graph of a non-zero constant polynomial, $p(x) = c$.) No zeroes.

(ii) The curve starts below the $x$-axis on the left, rises to a hump that stays below the axis, dips to a lower trough just to the right of the $y$-axis, and then climbs steeply, crossing the positive $x$-axis once. The hump is where students hesitate: it never reaches the axis, so it contributes nothing. One zero. The two turns make it look like a cubic, and a cubic can have fewer than three zeroes.

(iii) The curve rises from below on the left and crosses the negative $x$-axis, peaks above the axis near the $y$-axis, comes down and crosses the positive $x$-axis, dips a little below the axis, and crosses back up before rising again. That small dip is what makes the count three rather than two — the curve goes through the axis twice around it. Three zeroes.

(iv) A U-shaped curve lying entirely to the left of the $y$-axis. It comes down, crosses the negative $x$-axis, dips below it, and crosses it again on the way up. Both zeroes are negative, which is perfectly allowed. Two zeroes.

(v) A wave that crosses the $x$-axis four times: once far to the left, once just left of the origin, once just right of the origin (the trough between them lies below the axis, near the $y$-axis), and once far to the right after the tallest hump. Four zeroes.

(vi) The curve rises from below and crosses the negative $x$-axis once, forms a small hump, and comes back down to touch the $x$-axis — meeting it and turning up again without crossing. It then rises to a hump over the $y$-axis, comes down to touch the positive $x$-axis in the same way, and rises. That is one crossing and two touching points, each of which has $y = 0$. Three zeroes.

Common mistakes

• Part (i), answering $1$. The line crosses the $y$-axis, but that point is the value $p(0)$, not a zero. A zero is where the graph meets the $x$-axis.
• Part (ii), counting the hump. The left-hand hump comes close to the $x$-axis but stays below it. Only an actual meeting point counts, so the answer is $1$.
• Part (ii), assuming a cubic must have three zeroes. The degree gives an upper limit, not the exact number. This graph has two turns, like a cubic, and only one zero.
• Part (iii), missing the dip. Read carelessly, the trough on the right looks as if it sits on the axis. It goes below, giving two crossings close together — three zeroes in all, not two.
• Part (iv), thinking negative zeroes do not count. Both crossings are to the left of the origin; they are zeroes all the same.
• Part (vi), counting only crossings. The curve crosses once but touches twice. A touching point has $y = 0$, so it is a zero; leaving the touches out gives $1$ instead of $3$.

Practise next

• Exercise 2.2 — finding the zeroes of quadratic polynomials algebraically and checking them against the coefficients.
• Chapter 3, Exercise 3.1 — where graphs return: the solution of a pair of linear equations is the point where their two lines meet.