Polynomials is an important mathematics topic for school exams, competitive exams, entrance tests and aptitude preparation. It is useful because many questions can be solved quickly when the definitions, formulas and examples are understood properly.
This tutorial explains polynomials in simple language. It includes meaning, important terms, formulas, step-by-step methods, shortcut ideas, solved examples, practice questions, answers, common mistakes and FAQs.
The best way to learn polynomials is to first understand the rule, then solve easy examples, and then try exam-style questions. This page follows the same order.
What Is Polynomials?
A polynomial is an algebraic expression made of variables, coefficients and non-negative integer powers of variables.
Important Terms
| Point | Details |
|---|---|
| Term | Part of expression separated by plus or minus |
| Coefficient | Number multiplied with variable |
| Degree | Highest power of variable |
| Constant | Term without variable |
| Zero of polynomial | Value where polynomial becomes zero |
Important Formulas and Rules
| Point | Details |
|---|---|
| Linear polynomial | ax + b |
| Quadratic polynomial | ax^2 + bx + c |
| Cubic polynomial | ax^3 + bx^2 + cx + d |
| Remainder theorem | Remainder is p(a) when divided by x-a |
| Factor theorem | x-a is factor if p(a)=0 |
Step-by-Step Method
- Identify all terms.
- Find highest power for degree.
- Combine like terms.
- Use operations carefully.
- For zeros, put polynomial equal to zero.
Shortcut Ideas
- Only non-negative integer powers are allowed.
- Degree is highest variable power.
- Like terms can be added.
- A constant polynomial has degree 0 if non-zero.
- Use factor theorem to test factors.
Common Question Types
| Point | Details |
|---|---|
| Monomial | One term |
| Binomial | Two terms |
| Trinomial | Three terms |
| Linear | Degree one |
| Quadratic | Degree two |
Solved Examples
Example 1
Degree of 3x^2 + 2x + 1.
Degree is 2.
Example 2
Is x^-1 + 2 a polynomial?
No, negative power is not allowed.
Example 3
Add 2x + 3 and 4x + 5.
6x + 8.
Example 4
Find p(2) if p(x)=x^2+1.
p(2)=5.
Example 5
Is x-3 factor of x^2-9?
p(3)=0, so yes.
Exam Style Examples
Question 1
Find zero of x – 7.
x – 7 = 0 gives x = 7.
Zero makes polynomial equal to zero.
Question 2
Divide x^2 – 5x + 6 by x – 2. Remainder?
p(2)=4 – 10 + 6 = 0.
Use remainder theorem.
Question 3
Classify 5x^3 + 2x.
It has two terms and degree 3, so it is a cubic binomial.
Type can be by terms or degree.
Common Mistakes
- Calling expressions with negative powers polynomials.
- Taking number of terms as degree.
- Not combining like terms.
- Forgetting constant term.
- Confusing zero and coefficient.
Practice Questions
- Degree of x^3+2x.
- Degree of 5.
- Terms in x^2+3x+2.
- Add x+1 and 2x+3.
- Subtract x from 3x.
- p(1) for x^2+2.
- Is x^1/2 polynomial?
- Zero of x-4.
- Is x+2 factor of x^2-4?
- Coefficient of x in 5x+7.
Answers to Practice Questions
- 3
- 0
- 3
- 3x+4
- 2x
- 3
- No
- 4
- Yes
- 5
Frequently Asked Questions
What is polynomial?
It is an expression with variables and non-negative integer powers.
What is degree?
Highest power of variable.
What is monomial?
Polynomial with one term.
What is zero of polynomial?
Value that makes polynomial equal to zero.
What is factor theorem?
x-a is factor if p(a)=0.
Revision Notes
Polynomials are built from terms and powers. Degree is the highest power, not the number of terms.
For revision, keep a small formula list and solve a few mixed questions. Do not only read the solution. Try the question first, then compare your steps with the answer.
Conclusion
Polynomials become simple when terms, degree and operations are understood clearly.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Polynomials questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.