Chapter 3.3 — Exercise 3.2 — Graphical Representation
Graphical representation and finding zeroes using the graph. This is Lesson 3 of 5 in Chapter 3: Polynomials.
Where the Curve Meets the Axis
A polynomial's zeroes and its graph are two views of the same fact. Exercise 3.2 makes that connection precise: the zeroes of p(x) are exactly the x-coordinates of the points where the graph of y = p(x) crosses or touches the x-axis — true for a straight line, a parabola, or a curve of any higher degree, regardless of how complicated the polynomial itself looks.
The Linear Case
For p(x) = 2x − 5, setting p(x) = 0 gives x = 5/2 — and the graph of y = 2x − 5 crosses the x-axis at exactly (2.5, 0), the same value. In general, a linear polynomial ax + b (a ≠ 0) has exactly one zero, matching its single intersection with the x-axis. A straight line, unless it happens to run perfectly horizontal (which only happens when a = 0, disqualifying it from being genuinely linear), can never cross the x-axis more than once or avoid it entirely — its single, guaranteed intersection is what makes the linear case the simplest of the three covered in this exercise.
The Quadratic Case: Three Possible Shapes
The graph of a quadratic ax² + bx + c is a parabola — opening upward if a > 0, downward if a < 0 — and it can meet the x-axis in exactly three ways:
| Case | What happens | Number of zeroes |
|---|---|---|
| 1 | Parabola cuts the x-axis at two distinct points | 2 (distinct) |
| 2 | Parabola touches the x-axis at exactly one point | 1 (repeated) |
| 3 | Parabola stays entirely above or entirely below the x-axis | 0 |
x² − 5x + 6 factorises to (x − 2)(x − 3), giving zeroes at x = 2 and x = 3 — exactly the two points where y = x² − 5x + 6 crosses the x-axis, Case 1 in the table above.
The Cubic Case, and the General Pattern
A cubic's graph can cross the x-axis at one, two, or three points. y = x³ − 4x crosses at three distinct points (three zeroes); y = x³ touches the axis at only the origin (one zero); y = x³ − x² crosses at two points (two zeroes). The rule connecting degree to zero-count is simple and worth stating directly:
A polynomial of degree n has at most n zeroes — linear: at most 1, quadratic: at most 2, cubic: at most 3y = x³ is worth noting specifically: its graph doesn't cross the x-axis cleanly like x³ − 4x does, but instead flattens out and slides through the origin, still counting as exactly one zero rather than a repeated one. Not every cubic's behaviour near a zero looks identical, even when the zero-count comes out the same.
Six Graphs, Zeroes Counted at a Glance
The exercise opens with six graphs of unnamed polynomials p(x), asking only for the number of zeroes visible in each — a pure reading exercise, no algebra required:
| Graph | Behaviour | Zeroes |
|---|---|---|
| (i) | Never touches the x-axis | 0 |
| (ii) | Touches the x-axis at one point | 1 |
| (iii) | Crosses the x-axis at three points | 3 |
| (iv) | Crosses the x-axis at two points | 2 |
| (v) | Crosses the x-axis at four points | 4 |
| (vi) | Crosses the x-axis at three points | 3 |
Finding Zeroes Algebraically First
Before plotting, the exercise asks for zeroes found by solving p(x) = 0 directly:
- p(x) = 3x → 3x = 0 → x = 0
- p(x) = x² + 5x + 6 → splits to (x + 3)(x + 2) = 0 → x = −3 or x = −2
- p(x) = (x + 2)(x + 3) — already factored, same zeroes as above: x = −2 or x = −3
- p(x) = x⁴ − 16 → factorises via difference of squares to (x − 2)(x + 2)(x² + 4) = 0; since x² + 4 can never be zero for real x, the only real zeroes are x = 2 and x = −2
Plotting a Parabola, Point by Point
For p(x) = x² − x − 12, building a table of values across a range of x, then plotting and joining the points, traces out the full parabola:
| x | −4 | −3 | 0 | 2 | 4 | 5 |
|---|---|---|---|---|---|---|
| y | 8 | 0 | −12 | −10 | 0 | 8 |
The graph crosses the x-axis at (−3, 0) and (4, 0), so the zeroes are x = −3 and x = 4 — and substituting both back into the original polynomial confirms it: p(−3) = 9 + 3 − 12 = 0, and p(4) = 16 − 4 − 12 = 0. This substitution-based check is called justification, and it's the standard way to confirm a graphically-read zero algebraically — a graph alone can suggest where a zero sits, but only substitution proves it exactly, since a hand-drawn curve can never be trusted down to the precise coordinate.
Four More Parabolas, Each a Different Case
Plotting four further quadratics shows all three cases from the table above actually occurring in practice, not just as an abstract possibility:
| Polynomial | Graph behaviour | Zeroes |
|---|---|---|
| x² − 6x + 9 | Touches the x-axis at exactly one point (3, 0) | x = 3 (repeated) |
| x² − 4x + 5 | Stays entirely above the x-axis | None |
| x² + 3x − 4 | Crosses at (−4, 0) and (1, 0) | x = −4 and x = 1 |
| x² − 1 | Crosses at (−1, 0) and (1, 0) | x = −1 and x = 1 |
x² − 6x + 9 is worth a second look: it factorises as (x − 3)², so both "zeroes" coincide at the same point, x = 3 — which is exactly why its graph only touches the axis rather than crossing through it, Case 2 from the earlier table. x² − 4x + 5, by contrast, has no real zeroes at all — trying to factorise it the usual way fails to produce two real numbers that multiply and add correctly, which is the algebraic signal matching what the graph shows directly: a parabola sitting entirely above the axis, Case 3.
Two Fractional Zeroes, Verified
The closing question asks why 1/4 and −1 are zeroes of p(x) = 4x² + 3x − 1. Substituting each directly settles it: p(1/4) = 4(1/16) + 3(1/4) − 1 = 1/4 + 3/4 − 1 = 0, and p(−1) = 4(1) + 3(−1) − 1 = 4 − 3 − 1 = 0. Both check out, confirming both are genuine zeroes — even though neither is a whole number.
From Pictures Back to Formulas
Reading zeroes off a graph is powerful, but slow for anything beyond quadratics — plotting six or more points by hand for every polynomial you meet doesn't scale well as a general method. Exercise 3.3 replaces the picture with a formula, connecting a polynomial's zeroes directly to its coefficients without needing a graph or even factorisation. For the substitution-based zero-checking this exercise builds on, see Exercise 3.1, or revisit the core definitions in the chapter introduction.