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Cubic polynomial and its zeros

A cubic polynomial is a polynomial of degree 3, usually written as ax^3+bx^2+cx+d where a is not zero. Its zeros are the values of x that make the polynomial equal to 0.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    A cubic polynomial is a polynomial of degree 3, usually written as ax^3+bx^2+cx+d where a is not zero. Its zeros are the values of x that make the polynomial equal to 0.

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete 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.

Definition

A cubic polynomial is a polynomial of degree 3, usually written as ax^3+bx^2+cx+d where a is not zero. Its zeros are the values of x that make the polynomial equal to 0.

Example

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

Rule to remember

For ax^3+bx^2+cx+d, the number of zeros is at most 3. If the polynomial factorises into linear factors, each factor gives a zero.

Memory hook

Cubic can give three x-axis hits.

Examples and method

Worked example

For x^3-4x^2-x+4, group terms: x^2(x-4)-1(x-4)=(x^2-1)(x-4)=(x-1)(x+1)(x-4). The zeros are 1, -1, and 4.

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

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.

Exam tip

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

Quick check

How many zeros can a cubic polynomial have at most?

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.

Answer writing and exam use

1-mark answer

A cubic polynomial is a polynomial of degree 3, usually written as ax^3+bx^2+cx+d where a is not zero. Its zeros are the values of x that make the polynomial equal to 0.

2-mark answer

A cubic polynomial is a polynomial of degree 3, usually written as ax^3+bx^2+cx+d where a is not zero. Its zeros are the values of x that make the polynomial equal to 0. For ax^3+bx^2+cx+d, the number of zeros is at most 3. If the polynomial factorises into linear factors, each factor gives a zero. For x^3-6x^2+11x-6, the zeros are 1, 2, and 3.

3-mark answer

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. For ax^3+bx^2+cx+d, the number of zeros is at most 3. If the polynomial factorises into linear factors, each factor gives a zero. For x^3-4x^2-x+4, group terms: x^2(x-4)-1(x-4)=(x^2-1)(x-4)=(x-1)(x+1)(x-4). The zeros are 1, -1, and 4. Find the zeros of a cubic polynomial, or use factorisation to verify a stated root. A common wrong answer is to give only one or two zeros after partial factorisation. A cubic needs complete factorisation before the final answer.
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