Cubic polynomial and its zeros Class 10 Maths | Polynomials

Understand it

A cubic polynomial can have up to three zeros. In many exam questions, students need to identify the zeros by factorising, by using a given factor, or by checking a graph. The important idea is that each zero makes the polynomial vanish at that x-value.

Example

For x³-6x²+11x-6, the zeros are 1, 2, and 3.

Formula and rule

For ax³+bx²+cx+d, the number of zeros is at most 3.

If the polynomial factorises into linear factors, each factor gives a zero.

See it worked out

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

Worked example

1For x³-4x²-x+4, group terms: x²(x-4)-1(x-4)=(x²-1)(x-4)=(x-1)(x+1)(x-4).
2The zeros are 1, -1, and 4.

Method you can reuse

1Set the polynomial equal to zero.
2Try grouping or factor theorem if a factor is known.
3Reduce it to linear factors.
4Solve each factor and list all zeros.

Visual tip

A cubic graph may cross the x-axis up to three times. Each crossing or touching point gives a zero.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

How many zeros can a cubic polynomial have at most?

Think first, then show answer

A cubic polynomial can have at most three zeros. In practice, it may have three real zeros, two real zeros with one repeated root, or fewer real zeros depending on the polynomial.

Common wrong answer

A common wrong answer is to give only one or two zeros after partial factorisation. A cubic needs complete factorisation before the final answer.

Don't mix this up

Students sometimes stop after finding two zeros and forget that a cubic polynomial may have three zeros.

Exam tip

After factorising a cubic polynomial, keep solving until the expression is fully broken into linear factors.

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 of a cubic polynomial, or use factorisation to verify a stated root.

1
Enough for 1 mark
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.
2
Add for 2 marks
Example: For x³-6x²+11x-6, the zeros are 1, 2, and 3.
3
Add for 3 marks
A cubic polynomial can have up to three zeros.

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

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.

Verification of given zeros

To verify whether a number is a zero of a polynomial, substitute that number into the polynomial and check whether the value becomes 0.

Division algorithm for polynomials

For polynomials p(x) and g(x), with g(x) not equal to zero, p(x)=g(x)q(x)+r(x), where the degree of r(x) is less than the degree of g(x).

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.

Previous: Quadratic polynomial and its zeros Next: Relationship between zeros and coefficients of a quadratic polynomial Back to full mind map

Cubic polynomial and its zeros

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Check yourself

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Cubic polynomial and its zeros

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