Polynomials-Exercise 2.1

Exercise 2.1 of NCERT Mathematics Chapter 2 introduces students to the basic definition and terminology of polynomials, focusing on recognizing polynomials, identifying their degrees, and understanding coefficients and terms. This exercise strengthens the foundation required for solving polynomial equations, recognizing zeros, and preparing for factorization and graphical analysis in higher classes. Students are encouraged to explore examples and classifications—such as linear, quadratic, and cubic polynomials—helping build essential algebraic thinking for success in exams and future studies.

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TRIGONOMETRIC FUNCTIONS-Exercise 3.2

Exercise • Jan 2026

Trigonometric Functions form a crucial foundation of higher mathematics and play a vital role in physics, engineering, astronomy, and real-life proble...

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Maths

TRIGONOMETRIC FUNCTIONS-Exercise 3.1

Exercise • Jan 2026

Trigonometric Functions form a crucial foundation of higher mathematics and play a vital role in physics, engineering, astronomy, and real-life proble...

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Exercise
October 13, 2025  |  By Academia Aeternum

Polynomials-Exercise 2.1

Maths - Exercise

Q1. The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

Graphs
Fig 2.10


  1. In figure (i), \(p(x)\) neither crosses nor touches the \(x\)-axis. Therefore, p(x) has no \(x\) for which \(p(x)\)=0.
  2. In figure (ii) \(px\) crosses \((x)\)-axis once. Therefore, there is exactly one real zero of \(p(x)\)
  3. In figure (iii) \(p(x)\) crosses \(x\)-axis three times. Thus \(p(x)\) has three distinct real zeros.
  4. In fig (iv) \(p(x)\) crosses \(x\)-axis two times. Therfore, there exist two zeros of \(p(x)\)
  5. In fig (v) \(p(x)\) crosses \(x\)-axis 4 times. Therfore, \(p(x)\) has four real zeros.
  6. In fig (vi) \(p(x)\) crosses \(x\)-axis once nd touches twice(i.e., has two points of tangency). Therefore, there are three real zeros of \(p(x)\)


Notes/Explanations:


  • When a graph crosses the \(x\)-axis, the zero is a simple root.
  • When it touches but does not cross the \(x\)-axis (tangent), it is a repeated root (even multiplicity).
  • The total number of times the graph crosses or touches the \(x\)-axis (counting multiplicity) gives the number of real zeros.

Frequently Asked Questions

A polynomial is an algebraic expression that combines variables and numbers, using only non-negative whole number exponents.

Polynomials are classified by their highest exponent: linear (power one), quadratic (power two), cubic (power three), and higher-degree polynomials.

The degree of a polynomial is the largest exponent of the variable found in the polynomial.

A linear polynomial is an expression with the variable raised to one, for example, "a times x plus b."

A quadratic polynomial includes the variable raised to the second power, like "a times x squared plus b times x plus c."

A cubic polynomial contains the variable raised to the third power, such as "a times x cubed plus b times x squared plus c times x plus d."

The coefficient is the number multiplied by the variable in each term, for example, in "four x squared," the number four is the coefficient.

You add polynomials by merging terms that have the same variables and powers, using regular addition for their coefficients.

Subtracting polynomials means you subtract the coefficients of terms that have matching variables and exponents.

To multiply polynomials, multiply every term in one polynomial by every term in the other and then add any like terms.

The zero of a polynomial is a value for the variable that makes the whole expression equal to zero.

The Factor Theorem says if a polynomial equals zero when you substitute a number for the variable, then the expression "variable minus that number" is a factor of the polynomial.

The Remainder Theorem tells us that if you divide a polynomial by "variable minus a number," the remainder is what you get when you plug that number into the polynomial.

To factorize a polynomial, rewrite it as a multiplication of simpler polynomials, just like splitting a number into its factors.

Polynomials are crucial because they help in describing patterns, solving equations, and modeling real-life scenarios in mathematics and science.

Polynomials help to describe curves, solve equations, explain scientific laws, and are widely used in fields such as engineering, physics, and economics.

Quadratic equations can be solved by finding their factors, setting each factor equal to zero, or using the quadratic formula to find the values of the variable.

No, all exponents in a polynomial must be zero or positive whole numbers; negative or fractional powers are not allowed.

The standard form of a polynomial lists the terms in order from the highest exponent to the lowest.

A constant polynomial is just a number with no variable; its degree is zero.

Like terms are the ones that have the same variables raised to the same exponents; only their coefficients are different.

You find the value by replacing the variable with the given number and calculating the result using arithmetic.

The sum of the zeroes of a quadratic polynomial, "a x squared plus b x plus c," is equal to minus b divided by a.

The product of the zeroes of a quadratic polynomial, "a x squared plus b x plus c," is equal to c divided by a.

Polynomials are graphed to show how the output value changes when the variable changes; the degree of the polynomial determines the complexity of the graph's shape.

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