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351. Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also, verify the relationship between the zeroes and coefficients in each case:
x³ - 4x² + 5x - 2; 2, 1, 1

Let p(x) = x³ - 4x² + 5x - 2
Comparing the given polynomial with $ax cubed plus bx squared plus cx plus straight d comma$ we get
a = 1, b = -4, c = 5 and d = -2
Now, $straight p left parenthesis 2 right parenthesis space equals space left parenthesis 2 right parenthesis cubed minus 4 left parenthesis 2 right parenthesis squared plus 5 cross times left parenthesis 2 right parenthesis minus 2 space space space space space space space space space equals 8 minus 16 plus 10 minus 2 space equals space 18 minus 18 space equals space 0$
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∴ 2, 1, 1 are the zeroes of x³ - 4x² + 5x - 2.
Hence Verified
Here, we have
$space space space space straight alpha equals 2 comma space straight beta equals 1 space and space straight gamma equals 1$
Now, $straight alpha plus straight beta plus straight gamma space equals space 2 plus 1 plus 1 equals 4 equals fraction numerator negative left parenthesis negative 4 right parenthesis over denominator 1 end fraction equals fraction numerator negative straight b over denominator straight a end fraction$
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 $equals 2 plus 1 plus 2 equals 5 over 1 equals straight c over straight a$
and $αβγ equals 2 cross times 1 cross times 1 equals 2 equals fraction numerator negative left parenthesis negative 2 right parenthesis over denominator 1 end fraction equals fraction numerator negative straight d over denominator straight a end fraction$
Hence verified.

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Short Answer Type

352. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, -7, -14 respectively.

171 Views

Long Answer Type

353. If the zeroes of the polynomial x³ – 3x²+ x + 1 are (a – b), a, (a + b), find a and b.

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354. If two zeroes of the polynomial x⁴ – 26x³-26x² + 138x –35 are 2 ±

square root of 3

, find other two zeroes.

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355. If the polynomial x⁴ – 6x³ + 16x² – 25x + 10 is divided by another polynomial x² - 2x + k, the remainder comes out to be x + a, find ‘k’ and ‘a’.

2902 Views