Look at the x-axis
Zeroes are where the graph meets the x-axis.
Where is a zero?
We know −1 and 4 are the zeroes of x² − 3x − 4. Its graph, y = x² − 3x − 4, is a curve.
Where on the picture do −1 and 4 show up?
What this lesson covers
The idea
The zeroes of p(x) are the x-coordinates of points where the graph of y = p(x) meets the x-axis; the line y = ax + b meets it only at (–b/a, 0), so ax + b has one zero.
Where is a zero?
We know −1 and 4 are the zeroes of x² − 3x − 4. Its graph, y = x² − 3x − 4, is a curve.
Where on the picture do −1 and 4 show up?
- On the x-axis, at x = −1 and x = 4
- On the y-axis
- At the lowest point of the curve
Slide the dot
First the straight line y = 2x + 3. Slide the dot until it sits on the x-axis.
Now the curve y = x² − 3x − 4. Find every place where it meets the x-axis.
Zeroes on the axis
The zeroes of a polynomial p(x) are precisely the x-coordinates of the points where the graph of y = p(x) meets the x-axis.
The line y = ax + b (a ≠ 0) meets the x-axis at exactly one point: (−b/a, 0) So a linear polynomial has exactly one zero.
For y = 2x + 3 the point is (−3/2, 0). For y = x² − 3x − 4 the points are (−1, 0) and (4, 0).
Notes
The zeroes of p(x) are the x-coordinates of the points where the graph of y = p(x) meets the x-axis. A line ax + b meets it at just one point, so it has one zero.
Check yourself
The graph of y = p(x) meets the x-axis at the points (5, 0) and (−2, 0). What are the zeroes of p(x)?
The line y = 2x − 6 meets the x-axis at the point (k, 0). Find k.
Answer: 3
k is the zero of 2x − 6: −b/a = 6/2 = 3. The line meets the x-axis at (3, 0).
How many zeroes does a linear polynomial ax + b (a ≠ 0) have?
The graph of p(x) never touches the x-axis. What does that tell us?
- 5 and −2 — correct. Yes! The zeroes are the x-coordinates of those points.
- 0 and 0. The y-coordinates are 0. The zeroes are the x-coordinates.
- 3. Nothing says to add them. The zeroes are 5 and −2.
- Exactly one — correct. Yes! Its graph is a straight line that meets the x-axis at exactly one point.
- Two. A straight line cannot cross the x-axis twice.
- None. Since a ≠ 0 the line is not flat, so it does meet the x-axis, at (−b/a, 0).
- p(x) has no zero — correct. Yes! Zeroes are where the graph meets the x-axis.
- p(x) has a zero at x = 0. x = 0 would mean the graph meets the x-axis at (0, 0). It never meets it.
- p(x) has one zero. One zero would mean one meeting point. There is none.