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Polynomials in One Variable Quiz - CBSE Class 9 Mathematics

Free practice quiz: Polynomials in One Variable (CBSE Class 9 Mathematics). 17 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This CBSE Class 9 Mathematics practice quiz has 17 multiple-choice questions on Polynomials in One Variable, and 17 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

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What Makes a Polynomial

  • A polynomial in one variable has terms like a x^n, where every power n is a whole number.
  • Expressions with x under a root, x in a denominator, or negative powers of x are not polynomials.
  • Each term has a coefficient. The term without x is the constant term. A missing power has coefficient zero.
General form
\[p(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0,\quad a_n\neq 0\]
Constant term
\[a_0\]
Product of powers (for expanding)
\[x^m\cdot x^n=x^{m+n}\]

Common mistake: Calling \sqrt{x}+a or \frac{1}{x}+b a polynomial. The powers there are a half and negative one, which are not whole numbers.

Degree, Leading Coefficient and Types

  • The degree is the highest power of x whose coefficient is not zero.
  • The leading coefficient is the number multiplying that highest power term, including its sign.
  • Polynomials are named by degree: linear, quadratic, cubic. They are also named by number of terms: monomial, binomial, trinomial.
Degree of p(x)
\[\deg p(x)=n \text{ when } a_n\neq 0\]
Linear form
\[ax+b,\ a\neq 0\]
Quadratic form
\[ax^2+bx+c,\ a\neq 0\]
Cubic form
\[ax^3+bx^2+cx+d,\ a\neq 0\]

Common mistake: Taking the coefficient of the first written term as leading. Always pick the coefficient of the highest power, even if terms are jumbled.

Value of a Polynomial

  • To find p(a), replace every x with a and simplify. Put negative values inside brackets.
  • Putting x equal to zero leaves only the constant term. Putting x equal to one adds all the coefficients.
Value at a
\[p(a)=a_na^n+\dots+a_1a+a_0\]
Sum of all coefficients
\[a_n+a_{n-1}+\dots+a_1+a_0\]

Common mistake: Writing -a^2 instead of (-a)^2. An even power of a negative number is positive.

Zeroes (Roots) of a Polynomial

  • A number c is a zero, or root, of p(x) if p(c) equals zero.
  • To find an unknown constant, substitute the given root and set the result equal to zero.
Condition for a zero
\[c \text{ is a zero of } p(x) \iff p(c)=0\]
Zero of a linear polynomial
\[ax+b=0 \Rightarrow x=-\frac{b}{a}\]

Common mistake: Setting p(c) equal to c or to one instead of zero when solving for an unknown constant.

Remainder and Factor Theorems

  • When p(x) is divided by x - a, the remainder equals p(a).
  • x - a is a factor of p(x) exactly when p(a) equals zero.
Remainder Theorem
\[p(x)\div(x-a)\ \Rightarrow\ \text{remainder}=p(a)\]
Factor Theorem
\[(x-a)\text{ is a factor of }p(x)\iff p(a)=0\]
Divisor x + a
\[p(x)\div(x+a)\ \Rightarrow\ \text{remainder}=p(-a)\]

Common mistake: Substituting x = a when dividing by x + a. The correct value to substitute is x = -a.

Exam tips

Beta: made with AI and checked by a second AI. A teacher has not reviewed it yet.

Sample questions

  1. Which of the following is a polynomial in one variable?

    Show answer

    Answer: A. 3x + 5

    A polynomial in one variable contains terms with only one variable. '3x + 5' has only the variable x.

  2. Which term is not present in the polynomial 4x^4 - x^2 + 6?

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    Answer: A. x^3

    The polynomial 4x^4 - x^2 + 6 does not have a term containing x^3.

  3. If x^2 - 5x + k is a polynomial, what is the value of k for x = 5 to be its root?

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    Answer: D. 25

    If x = 5 is a root, then 5^2 - 5(5) + k = 0. This gives k = 25.

  4. What is the sum of the coefficients of the polynomial 3x^4 - 2x^3 + x - 5?

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    Answer: A. -3

    Substitute x = 1: 3(1)^4 - 2(1)^3 + 1 - 5 = 3 - 2 + 1 - 5 = -3.

  5. Identify the type of polynomial: 4x + 7.

    Show answer

    Answer: B. Binomial

    A binomial is a polynomial with two terms. '4x + 7' consists of two terms, making it a binomial.

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More questions in this quiz

Answers and explanations are in the full quiz.

  1. Identify the degree of the polynomial 7x^3 - 4x + 2.
  2. What is the constant term in the polynomial 2x^2 + 3x + 5?
  3. For the polynomial p(x) = 5x^3 - 4x + 9, what is p(1)?
  4. Which expression represents 'the square of a polynomial'?
  5. What is the leading coefficient of the polynomial 3x^3 - 2x^2 + x - 7?
  6. Check if x - 2 is a factor of the polynomial p(x) = x^3 - 4x^2 + 6x - 8.
  7. If p(x) = x^2 + 2x + 4, what is the value of p(-2)?
  8. What is the value of p(0) for the polynomial p(x) = x^4 - 3x^3 + x - 2?
  9. In the polynomial 3x^3 - x^2 + 5x + 7, what is the term with the lowest degree?
  10. What is the polynomial obtained when x^2(3x^3 + x - 4) is expanded?
  11. If one of the roots of the polynomial px^2 + qx + r is 0, what is the relationship of p and r?
  12. Find the value of 'a' if the polynomial 3x^3 + ax^2 + 3x - 5 divided by x - 2 leaves a remainder 7.

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