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# 5 Common Mistakes in ICSE Class 10 Chapter Remainder And Factor Theorems

Although polynomials are essential to mathematics, many students make typical blunders when utilising them. The following article will discuss five common mistakes students make while using the Factor Theorem to solve polynomials, along with tips for avoiding them.

Let's first review the basics of Remainder and Factor Theorems for polynomials and the approaches used to answer the questions based on them before getting into the errors.

Overview of Remainder & Factor Theorems:

The very basic and initial thing that we need to understand is the factors, which means, a non-zero polynomial q(x) so that p(x) = q(x) g(x).

Factors then follow the Remainder Theorem that states: If p(x) is any polynomial of degree > 1 which is divided by (x – a) then the remainder is p(a), where ‘a’ is any real number.

Now, similarly, the Factor Theorem can be summarised as: If p(x) is any polynomial and ‘a’ is any real number then (x – a) is a factor of p(x) if and only if p(a) = 0.

After combining Remainder and Factor theorems, we have the following statement:

If p(x) is a polynomial of degree > 0, then it follows from the Remainder Theorem that:

• p(x) = (x – a) q(x) + p(a), where q(x) is a polynomial of degree (n – 1).
• If p(a) = 0 then p(x) = (x – a) q(x).
• If p(a) = 0, then (x – a) is a factor of p(x).

Let’s now focus on the 5 common mistakes made by the students.

Assume we have the following question:

“Find the value of ‘a’ if x – a is a factor of the polynomial 3x3 + x2 – ax – 81.”

Mistake 1

Students don’t apply the concept of Factor Theorem

If it is given that “(x – a) is a factor of the polynomial p(x)”.

We must apply the concept of the Factor Theorem, which says that “(x – a) is a factor of p(x) if and only if p(a) = 0”.

But, most of the students don’t apply this concept and follow some other method making the solving/calculation difficult or leading to an incorrect answer.

 Solution with mistake Correct Solution Can’t find the value of ‘a’.

Mistake 2

Assumed any random factor of the given polynomial

Students assume any random factor of the given polynomial leading to incorrect conclusions.

 Solution with mistake Correct Solution Though the value is correct, the approach is wrong. Because, if we take any other factor, we will obtain an incorrect value of ‘a’.

Mistake 3

Incorrectly calculated cube/cube root

 Solution with mistake Correct Solution If x – 2 is a factor of x3 – kx – 12, then the value of k is: Incorrectly taken 23 as 4 which gives an incorrect value of ‘k’. If x – 2 is a factor of x3 – kx – 12, then the value of k is: Find the value of ‘a’ if x – a is a factor of the polynomial 3x3 + x2 – ax – 81. The value of a3 is +ve, so ‘a’ must be +ve. Find the value of ‘a’ if x – a is a factor of the polynomial 3x3 + x2 – ax – 81.

Mistake 4

Using the Trial and Error method for Factorisation

For factorisation of polynomials, instead of applying the Factor theorem, students use the trial and error method which may lead to lengthy solutions or no conclusion for the factors.

 Solution with mistake Correct Solution So, (x + 1) is not a factor. This makes the calculation lengthy as we need to check for more factors.

Mistake 5

Writing factors with the comma in the final answer for Factorisation

 Solution with mistake Correct Solution When the factors are written with a comma, we’ll not get the actual polynomial, p(x) in this case.

Conclusion:

These were the five common mistakes observed among students while solving questions based on polynomials. To avoid losing marks in the examination, steer clear of these errors. It is advisable to meticulously read and analyse the question before deciding whether to apply the factor theorem or any other method, especially for subjective questions. When dealing with objective questions, opt for the simplest method, but be cautious of mistake 3. Therefore, strive to minimize errors by practising more questions. Stay tuned with TopperLearning for upcoming blogs on similar topics.

Understand the concepts through Revision Notes and Videos. Practice a variety of questions like Multiple Choice Questions, Subjective Questions and Assertion & Reasoning Questions. You can also use Sample Papers to get an idea of the actual paper. All of these resources are available for Maharashtra, CBSE and ICSE Boards students at TopperLearning.

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