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Geometrical meaning of zero of a polynomial

A zero of a polynomial is the value of x for which the polynomial becomes 0. On the graph, it is the x-coordinate where the curve touches or cuts the x-axis.

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Student-friendly explanation

If p(x)=0 at some value of x, that value is called a zero. Geometrically, zeros tell us where the graph meets the x-axis. This is why zeros are also called x-intercepts. For students, the key idea is simple: algebra gives the value, and the graph shows the same value as a point on the x-axis.

How to write this in exams

  1. 1

    Start with the exact idea

    A zero of a polynomial is the value of x for which the polynomial becomes 0. On the graph, it is the x-coordinate where the curve touches or cuts the x-axis.

  2. 2

    Then show how to use it

    1. Put p(x)=0 and solve if the polynomial is given. 2. If a graph is given, find x-coordinates where the graph meets the x-axis. 3. Write the values as zeros. 4. Mention that these are x-intercepts.

  3. 3

    Add one concrete example

    For p(x)=x-3, p(3)=0. So 3 is a zero, and the graph cuts the x-axis at x=3.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to read the y-coordinate instead of the x-coordinate. That is wrong because zeros are found where the graph crosses the x-axis.

Definition

A zero of a polynomial is the value of x for which the polynomial becomes 0. On the graph, it is the x-coordinate where the curve touches or cuts the x-axis.

Example

For p(x)=x-3, p(3)=0. So 3 is a zero, and the graph cuts the x-axis at x=3.

Rule to remember

If p(a)=0, then a is a zero of the polynomial p(x). On the graph of y=p(x), zeros are the x-intercepts.

Memory hook

Zero means x-value at the x-axis.

Examples and method

Worked example

For p(x)=x^2-1, p(1)=0 and p(-1)=0. So the zeros are 1 and -1. On the graph, the curve meets the x-axis at x=1 and x=-1.

Method to apply

1. Put p(x)=0 and solve if the polynomial is given. 2. If a graph is given, find x-coordinates where the graph meets the x-axis. 3. Write the values as zeros. 4. Mention that these are x-intercepts.

Diagram support

Draw the x-axis and mark where the polynomial graph crosses or touches it. Those points show the zeros clearly.

How CBSE asks it

Find the zeros from an expression or from a graph, and state their geometrical meaning in one line.

Avoid common mistakes

Common confusion

Many students think a zero means the y-value is zero for every point. Actually, it is the x-value that makes the polynomial equal to zero.

Common wrong answer

A common wrong answer is to read the y-coordinate instead of the x-coordinate. That is wrong because zeros are found where the graph crosses the x-axis.

Exam tip

When you see a graph, count the points where it meets the x-axis. Those x-values are the zeros.

Quick check

What does the zero of a polynomial tell us on its graph?

A zero of a polynomial tells us the x-value where the graph meets the x-axis, because at that point the polynomial value becomes zero.

Answer writing and exam use

1-mark answer

A zero of a polynomial is the value of x for which the polynomial becomes 0. On the graph, it is the x-coordinate where the curve touches or cuts the x-axis.

2-mark answer

A zero of a polynomial is the value of x for which the polynomial becomes 0. On the graph, it is the x-coordinate where the curve touches or cuts the x-axis. If p(a)=0, then a is a zero of the polynomial p(x). On the graph of y=p(x), zeros are the x-intercepts. For p(x)=x-3, p(3)=0. So 3 is a zero, and the graph cuts the x-axis at x=3.

3-mark answer

If p(x)=0 at some value of x, that value is called a zero. Geometrically, zeros tell us where the graph meets the x-axis. This is why zeros are also called x-intercepts. For students, the key idea is simple: algebra gives the value, and the graph shows the same value as a point on the x-axis. If p(a)=0, then a is a zero of the polynomial p(x). On the graph of y=p(x), zeros are the x-intercepts. For p(x)=x^2-1, p(1)=0 and p(-1)=0. So the zeros are 1 and -1. On the graph, the curve meets the x-axis at x=1 and x=-1. Find the zeros from an expression or from a graph, and state their geometrical meaning in one line. A common wrong answer is to read the y-coordinate instead of the x-coordinate. That is wrong because zeros are found where the graph crosses the x-axis.
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