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Chapter 2

Polynomials

NCERT solutions and explanations for Class 10 Mathematics Chapter 2 Polynomials — covering Degree of polynomials, zeroes, relation between zeroes and coefficients, and division algorithm for polynomials.

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Questions

6
Q1

Define the degree of a polynomial. Distinguish between linear, quadratic, and cubic polynomials.

The degree of a polynomial is the highest power of the variable with a non-zero coefficient. Degree 11 is linear, degree 22 is quadratic, and degree 33 is cubic.
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Q2

What is a zero of a polynomial? Find the zero of p(x)=2x−6p(x)=2x-6.

A zero of a polynomial p(x)p(x) is a value of xx for which p(x)=0p(x)=0. For 2x−6=02x-6=0, x=3x=3, so the zero is 33.
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Q3

State the relationship between zeroes and coefficients of a quadratic polynomial.

For p(x)=ax2+bx+cp(x)=ax^2+bx+c, a≠0a\ne0, if α\alpha and β\beta are zeroes, then α+β=−ba\alpha+\beta=-\frac ba and αβ=ca\alpha\beta=\frac ca.
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Q4

Find the zeroes of x2−5x+6x^2-5x+6 and verify the coefficient relations.

Factorise: x2−5x+6=(x−2)(x−3)x^2-5x+6=(x-2)(x-3). Zeroes are 22 and 33. Sum =5=−−51=5=-\frac{-5}{1} and product =6=61=6=\frac61.
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Q5

Form a quadratic polynomial whose zeroes are 44 and −1-1.

A quadratic with zeroes α\alpha and β\beta is k[x2−(α+β)x+αβ]k[x^2-(\alpha+\beta)x+\alpha\beta]. Taking k=1k=1, zeroes 4,−14,-1 give x2−3x−4x^2-3x-4.
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Q6

State the division algorithm for polynomials and explain the meaning of quotient and remainder.

If p(x)p(x) and g(x)g(x) are polynomials with g(x)≠0g(x)\ne0, then p(x)=g(x)q(x)+r(x)p(x)=g(x)q(x)+r(x), where q(x)q(x) is the quotient and either r(x)=0r(x)=0 or degree of r(x)r(x) is less than degree of g(x)g(x).
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