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Polynomials

Question

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Prove that $left parenthesis 3 minus square root of 7 right parenthesis squared$ is an irrational number.

Solution

left parenthesis 3 minus square root of 7 right parenthesis squared space space space space space space equals space 9 plus 7 minus 2.3. square root of 7 space space space space space space space equals 16 minus 6 square root of 7 space space space space space space

6 square root of 7

is an irrational number as 6 is a non-zero rational number and

square root of 7

is an irrational number and the product of a non-zero rational number with an irrational number is irrational.

Again, 16 is a rational number and

6 square root of 7

is an irrational number. So,

16 minus 6 square root of 7

is an irrational number since the difference of a rational number and an irrational number is irrational.

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