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Polynomials

Short Answer Type

341. Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following :
p(x) = x⁴ – 5x + 6, g(x) = 2 – x².

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342. Check whether the first polynomial is a factor of the second polynomial by applying the division algorithm.
t² – 3, 2t⁴ + 3t³ – 2t²–9t – 12

Here, the remainder is zero, hence t2 – 3 is a factor of 2r4 + 3t

Here, the remainder is zero, hence t² – 3 is a factor of 2r⁴ + 3t³ – 2t² – 9t – 12.

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343. Check whether the first polynomial is a factor of the second polynomial by applying the division algorithm.
x² + 3x + 1, 3x⁴ + 5x³ – 7x² + 2x + 2

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344. Check whether the first polynomial is a factor of the second polynomial by applying the division algorithm.
x³ – 3x + 1, x⁵ –4x³ + x² + 3x + 1.

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Long Answer Type

345. Obtain all other zeroes of 3x⁴ + 6x³ – 2x²- 10x - 5, if two of its zerores are

square root of 5 over 3 end root space and space minus square root of 5 over 3 end root

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346. On dividing x³ – 3x² + x + 2 by a polynomial g(x), the quotient and remainder, are x – 2 and –2x + 4, respectively. Find g(x).

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Short Answer Type

347. Give examples of polynomials p(x),g(x), q(x) and r(x), which satisfy the division algorithm and
deg p(x) = deg q(x)

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348. Give examples of polynomials p(x),g(x), q(x) and r(x), which satisfy the division algorithm and
deg r(x) = 0

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349. Give examples of polynomials p(x),g(x), q(x) and r(x), which satisfy the division algorithm and
deg q(x) = deg r(x)

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Long Answer Type

350. 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:
2x³+ x²–5x + 2; 1/2, 1,–2

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