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Mathematics · Polynomials · NCERT Examples

Example 5

A complete, independently verified solution for NCERT Example 5.

Mathematics · Chapter 2 · NCERT Examples

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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.