Exercise 2.4 — Factor Theorem

The Factor Theorem: A Key Tool for Factorisation

Exercise 2.4 of Class 9 Mathematics Chapter 2, Polynomials and Factorisation, introduces the Factor Theorem, which builds directly on the Remainder Theorem covered in the previous section. This theorem provides a quick way to check whether a linear polynomial is a factor of a larger polynomial, without performing long division. It is one of the most frequently tested concepts in CBSE, Telangana, and Andhra Pradesh board exams, since it forms the basis for factorising cubic and higher-degree polynomials.

Statement of the Factor Theorem

The Factor Theorem states that for a polynomial p(x) of degree one or more, and any real number a: (x − a) is a factor of p(x) if and only if p(a) = 0.

This follows directly from the Remainder Theorem, since the remainder when p(x) is divided by (x − a) is p(a). If that remainder is zero, the division is exact, which means (x − a) divides p(x) completely with no leftover term.

A Quick Shortcut: The (x + 1) and (x − 1) Tests

Exercise 2.4 highlights two especially useful shortcuts that save time during exams:

• (x − 1) is a factor of a polynomial if and only if the sum of all its coefficients equals zero. This is because p(1) = 0 simply adds up every coefficient.
• (x + 1) is a factor of a polynomial if and only if the sum of the coefficients of the even-power terms equals the sum of the coefficients of the odd-power terms. This comes from evaluating p(−1), where alternating signs separate the terms by parity of power.

For example, in x³ − x² − x + 1, the even-power coefficients (constant and x² terms) sum to −1 + 1 = 0, and the odd-power coefficients (x³ and x terms) sum to 1 − 1 = 0. Since both sums are equal, (x + 1) is a factor — confirmed by checking p(−1) = 0.

Using the Factor Theorem to Verify Multiple Factors

Several problems ask students to verify that two or three given linear expressions are all factors of a cubic polynomial. The method is the same each time: substitute the zero of each linear factor into the polynomial and check that the result is zero. For example, to show that (x − 2), (x + 3), and (x − 4) are factors of x³ − 3x² − 10x + 24, students calculate p(2), p(−3), and p(4) separately — each substitution must simplify to zero.

This kind of question reinforces careful substitution with negative numbers and prepares students for the full factorisation problems that follow.

Complete Factorisation Using the Factor Theorem

The most important question type in this exercise asks students to factorise a cubic polynomial completely. The general strategy involves three steps:

• Use trial values (usually ±1, ±2, and divisors of the constant term) to find one zero of the polynomial using the Factor Theorem.
• Divide the cubic polynomial by the linear factor found, using long division, to get a quadratic quotient.
• Factorise the resulting quadratic by splitting the middle term, giving the complete factorisation as a product of three linear factors.

For example, for x³ − 2x² − x + 2, both x = 1 and x = −1 make the polynomial zero, so (x − 1) and (x + 1) are factors. Dividing by (x² − 1) gives a quotient of (x − 2), so the full factorisation is (x − 1)(x + 1)(x − 2).

Proof-Based and Common Factor Problems

Some of the harder problems in Exercise 2.4 ask students to prove general relationships, such as showing that if x² − 1 is a factor of a quartic polynomial, then the sum of the coefficients of even-power terms equals the sum of the odd-power terms, and both equal zero. Other problems involve two polynomials sharing a common linear factor — students set up equations using f(−a) = 0 and g(−a) = 0 for both polynomials and solve simultaneously to find unknown constants or relationships between them.

Common Mistakes to Avoid

• Forgetting to check the sign of the zero — (x + a) gives x = −a, not x = a.
• Errors while splitting the middle term during the final quadratic factorisation step.
• Skipping verification — always check that the product of the final factors gives back the original polynomial.
• Confusing the Remainder Theorem (finds the remainder) with the Factor Theorem (confirms a factor when the remainder is zero).

What This Lesson Prepares You For

Mastering the Factor Theorem is essential for solving polynomial equations and is a recurring topic in CBSE, Telangana, and Andhra Pradesh board exams. Students should revisit Exercise 2.3 on the Remainder Theorem to strengthen the foundation for this lesson, and continue practising factorisation techniques that will be applied throughout the rest of this chapter and in solving quadratic and cubic equations in higher classes.