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Mathematics · Polynomials

NCERT Examples

All 5 worked examples from Polynomials, with independently verified solutions and direct navigation.

Mathematics · Chapter 2 · NCERT Examples

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NCERT Example

Example 1

geometrical meaning of zeroes · polynomial graphs

Look at the graphs in Figure 2.9. Each is the graph of y = p(x), where p(x) is a polynomial. For each graph, find the number of zeroes of p(x).

Six polynomial graphs labelled i to vi, showing one, two, three, one, one and four x-axis intersections respectively.
Figure 2.9: six graphs of y = p(x).

Solution

  1. The graph intersects the x-axis once, so it has 1 zero.

  2. The graph intersects the x-axis twice, so it has 2 zeroes.

  3. The graph intersects the x-axis three times, so it has 3 zeroes.

  4. The graph intersects the x-axis once, so it has 1 zero.

  5. The graph touches the x-axis once, so it has 1 zero.

  6. The graph intersects the x-axis four times, so it has 4 zeroes.

Final answer

The numbers of zeroes are 1, 2, 3, 1, 1 and 4 respectively.

NCERT Example

Example 2

quadratic polynomials · zeroes and coefficients

Find the zeroes of the quadratic polynomial x² + 7x + 10, and verify the relationship between the zeroes and the coefficients.

Solution

x2+7x+10=(x+2)(x+5)x^2+7x+10=(x+2)(x+5)

Thus the zeroes are −2 and −5.

(−2)+(−5)=−7=−ba(-2)+(-5)=-7=-\frac{b}{a}
(−2)(−5)=10=ca(-2)(-5)=10=\frac{c}{a}

Final answer

The zeroes are −2 and −5; their sum is −7 and product is 10.

NCERT Example

Example 3

quadratic polynomials · zeroes and coefficients

Find the zeroes of the polynomial x² − 3 and verify the relationship between the zeroes and the coefficients.

Solution

x2−3=(x−3)(x+3)x^2-3=(x-\sqrt3)(x+\sqrt3)

The zeroes are √3 and −√3.

3−3=0=−ba\sqrt3-\sqrt3=0=-\frac{b}{a}
(3)(−3)=−3=ca(\sqrt3)(-\sqrt3)=-3=\frac{c}{a}

Final answer

The zeroes are √3 and −√3; their sum is 0 and product is −3.

NCERT Example

Example 4

forming a quadratic polynomial · sum and product of zeroes

Find a quadratic polynomial whose zeroes have sum −3 and product 2.

Solution

x2−(α+β)x+αβ=x2+3x+2x^2-(\alpha+\beta)x+\alpha\beta=x^2+3x+2

Every non-zero real multiple of this polynomial has the same zeroes.

Final answer

x2+3x+2x^2+3x+2

NCERT Example

Example 5

cubic polynomials · zeroes and coefficients

Verify that 3, −1 and −1/3 are the zeroes of p(x) = 3x³ − 5x² − 11x − 3, and verify the relationship between the zeroes and the coefficients.

Solution

p(3)=81−45−33−3=0p(3)=81-45-33-3=0
p(−1)=−3−5+11−3=0p(-1)=-3-5+11-3=0
p(−13)=−19−59+113−3=0p\left(-\frac13\right)=-\frac19-\frac59+\frac{11}{3}-3=0
3−1−13=53=−ba3-1-\frac13=\frac53=-\frac ba
3(−1)+(−1)(−13)+(−13)3=−113=ca3(-1)+(-1)\left(-\frac13\right)+\left(-\frac13\right)3=-\frac{11}{3}=\frac ca
3(−1)(−13)=1=−da3(-1)\left(-\frac13\right)=1=-\frac da

Final answer

All three values are zeroes, and the sum, pairwise-product sum and product agree with the coefficient relationships.