(i) \(4x^2 – 3x + 7\)
(ii) \(y^2 + \sqrt{2}\)
(iii) \(3\sqrt{t} + t\sqrt{2}\)
(iv) \(y + \frac{2}{y}\)
(v) \(x^{10} + y^3 + t^{50}\)
📘 Concept & Theory Important Theory ›
A polynomial in one variable is an algebraic expression in which:
- Only one variable is present.
- The powers (exponents) of the variable are whole numbers \(0,1,2,3,\dots\)
- No variable should appear in the denominator.
- No variable should have negative or fractional exponents.
General form of a polynomial in one variable:
\[\small a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 \]where \(a_0,a_1,a_2,\dots\) are constants and \(n\) is a whole number.
🗺️ Solution Roadmap Step-by-step Plan ›
- Solution Roadmap
- Check the number of variables present.
- Check whether exponents of variables are whole numbers.
- Verify that no variable occurs in the denominator or under a root.
- Decide whether the expression satisfies the definition of a polynomial in one variable.
✏️ Solution Complete Solution ›
- Given expression: \[\small 4x^2-3x+7 \]
Count the variables.
Only one variable \(x\) is present.
Check powers of the variable.
\[\small x^2,\quad x^1,\quad x^0 \]- All exponents are whole numbers.
- Conclusion
- \[\small 4x^2-3x+7\]is a polynomial in one variable.
✏️ Solution Complete Solution ›
- Given expression: \[\small y^2+\sqrt{2} \]
Count the variables.
Only one variable \(y\) is present.
Check powers of the variable.
\[\small y^2 \]Exponent 2 is a whole number- Examine the constant term
- \[\small \sqrt{2}\]\(\sqrt{2}\) is an irrational number, but it is a constant. Constants are allowed in polynomials.
- conclusion
- \[\small y^2+\sqrt{2}\]is a polynomial in one variable.
✏️ Solution Complete Solution ›
- Given expression: \[\small 3\sqrt{t}+t\sqrt{2}\]
Count the variables.
Only one variable \(t\) is present.
Check powers of the variable.
\[\small t^1,\quad t^{1/2} \]Exponent 1 is a whole number, but 1/2 is not.- conclusion
- \[\small 3\sqrt{t}+t\sqrt{2}\]is not a polynomial
✏️ Solution Complete Solution ›
- Given expression: \[\small y+\frac{2}{y}\]
Count the variables.
Only one variable \(y\) is present.
Check powers of the variable.
\[\small y^1,\quad y^{-1} \]Exponent 1 is a positive whole number, but -1 is not. Negative exponents are not allowed in polynomials.- conclusion
- \[\small y+\frac{2}{y}\]is not a polynomial
✏️ Solution Complete Solution ›
- Given expression: \[\small x^{10}+y^3+t^{50}\]
Count the variables.
Three variables \(x,y,t\) are present. Polynomials in one variable cannot have more than one variable.
- conclusion
- \[\small x^{10}+y^3+t^{50}\]is not a polynomial because a polynomial in one variable must contain only one variable.
📊 Graph / Figure Graph / Figure ›
🎯 Exam Significance Exam Significance ›
- Identification of polynomials is the foundation of the entire chapter.
- Questions based on polynomial classification are frequently asked in school examinations and board objective papers.
-
Competitive entrance examinations test conceptual clarity about:
- whole number exponents,
- variables in denominators,
- fractional powers,
- number of variables.
- Strong understanding of this concept helps in later topics such as: factorisation, algebraic identities, graphs, and calculus.