‘रचना-प्रयोग’ का मुख्य उद्देश्य क्या है?

व्याकरण का व्यावहारिक प्रयोग सिखाना

‘द्वितीयः’ किस प्रकार की संख्या है?

क्रमवाचक (दूसरा)

Find the sum and product of the zeroes of \(x^2 - 5x + 6\).

Sum \(= -(-5)/1 = 5\); product \(= 6/1 = 6\) (the zeroes are 2 and 3).

What is the geometrical meaning of the zeroes of a polynomial?

They are the x-coordinates of the points where the graph of the polynomial intersects the x-axis.

What is the degree of a polynomial? Give the degree of a linear, quadratic and cubic polynomial.

The highest power of the variable in the polynomial; linear has degree 1, quadratic degree 2 and cubic degree 3.

What is the geometrical meaning of the zeroes of a polynomial?

They are the x-coordinates of the points where the graph of the polynomial cuts or touches the x-axis (refer to the graphs in your NCERT textbook).

For a quadratic polynomial \(ax^2 + bx + c\), write the sum and product of the zeroes.

Sum of the zeroes \(= -b/a\) and product of the zeroes \(= c/a\).

Find the zeroes of \(x^2 - 2x - 8\) and verify the relationship with the coefficients.

\(x^2 - 2x - 8 = (x - 4)(x + 2)\), so the zeroes are 4 and \(-2\); sum \(= 2 = -(-2)/1\) and product \(= -8 = -8/1\), which verifies the relations.

Form a quadratic polynomial whose zeroes are 3 and \(-2\).

Sum \(= 1\), product \(= -6\), so the polynomial is \(x^2 - (\text{sum})x + \text{product} = x^2 - x - 6\).

How many zeroes can a quadratic and a cubic polynomial have at most?

A quadratic polynomial has at most 2 zeroes and a cubic polynomial has at most 3 zeroes (equal to its degree).

If one zero of \(x^2 - 5x + k\) is 2, find \(k\).

Substituting \(x = 2\): \(4 - 10 + k = 0\), so \(k = 6\).

Write the relationship between the zeroes and coefficients of a cubic polynomial \(ax^3 + bx^2 + cx + d\).

Sum of zeroes \(= -b/a\), sum of products of zeroes taken two at a time \(= c/a\), and product of zeroes \(= -d/a\).

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CBSE Class 10 — Notes, Chapters & Practice Quizzes

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Chapter 2: PolynomialsClass 10 Mathematics — summary, notes, extra questions & NCERT solutions

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The zero of the polynomial p(x) = x − 3 is:

Summary

A polynomial is an algebraic expression in which the powers of the variable are whole numbers. This chapter focuses on the geometrical meaning of the zeroes of a polynomial — the points where its graph meets the x-axis — and on the relationship between the zeroes and the coefficients. For a quadratic polynomial \(ax^2 + bx + c\), the sum of the zeroes is \(-b/a\) and the product is \(c/a\); similar relations extend to cubic polynomials. These relationships let students form a polynomial from its zeroes and verify answers quickly, linking algebra with the shape of curves.

Geometrical meaning of zeroesRelationship between zeroes and coefficientsQuadratic and cubic polynomials

Key terms

Polynomial

An expression with non-negative integer powers of the variable.

Zero of a polynomial

A value of the variable for which the polynomial equals zero.

Degree

The highest power of the variable in the polynomial.

Sum/product of zeroes

For \(ax^2+bx+c\): sum \(= -b/a\), product \(= c/a\).

Extra questions & answers

EXERCISE 2.1

EXERCISE 2.2

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Graph the polynomial and spot its zeros

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y = a*x*x + b*x + c

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An expression with non-negative integer powers of the variable.

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