Geometrical meaning of zero of a polynomial Class 10 Maths | Polynomials

Understand it

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.

Example

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

Formula and rule

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.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

For p(x)=x²-1, p(1)=0 and p(-1)=0.

1So the zeros are 1 and -1.
2On the graph, the curve meets the x-axis at x=1 and x=-1.

Method you can reuse

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

Sketch it in your answer

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer

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.

Common wrong answer

A common wrong answer is to read the y-coordinate instead of the x-coordinate.

Why it loses marks: Zeros are found where the graph crosses the x-axis.

Don't mix this up

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.

Exam tip

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

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

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

1
Enough for 1 mark
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
Add for 2 marks
Example: For p(x)=x-3, p(3)=0. So 3 is a zero, and the graph cuts the x-axis at x=3.
3
Add for 3 marks
If p(x)=0 at some value of x, that value is called a zero.

Use the concept graph around this idea instead of revising it in isolation.

Linear polynomial and its zero

A linear polynomial has highest power 1, usually written as ax+b where a is not zero. Its zero is the value of x that makes ax+b equal to 0.

Quadratic polynomial and its zeros

A quadratic polynomial is a polynomial of degree 2, usually written as ax²+bx+c where a is not zero. Its zeros are the values of x that make the polynomial equal to 0.

Cubic polynomial and its zeros

A cubic polynomial is a polynomial of degree 3, usually written as ax³+bx²+cx+d where a is not zero. Its zeros are the values of x that make the polynomial equal to 0.

Sign pattern and graph interpretation

The sign pattern of a polynomial tells whether the graph lies above or below the x-axis in different intervals. The x-intercepts show the zeros, and the shape of the graph helps interpret the number of real zeros.

Next: Linear polynomial and its zero Back to full mind map

Geometrical meaning of zero of a polynomial

Understand it

See it worked out

Check yourself

Write it in the exam

Practice this concept with focused MCQs

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Geometrical meaning of zero of a polynomial

Understand it

See it worked out

Check yourself

Write it in the exam

Related concepts

Practice this concept with focused MCQs

Help improve this page