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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

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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.

128 Views

Long Answer Type

345. Obtain all other zeroes of 3x⁴ + 6x³ – 2x²- 10x - 5, if two of its zerores are $square root of 5 over 3 end root space and space minus square root of 5 over 3 end root$ .

Let $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ and $straight alpha space equals space square root of 5 over 3 end root comma space space space space straight beta space equals space minus square root of 5 over 3 end root$
∵ $square root of 5 over 3 end root space and space minus square root of 5 over 3 end root$ are zeroes of p(x), then
$open parentheses straight x minus square root of 5 over 3 end root close parentheses open parentheses straight x plus square root of 5 over 3 end root close parentheses equals open parentheses straight x squared minus 5 over 3 close parentheses$ is a factor of p(x).
Now by applying division algorithm:
Let and ∵ are zeroes of p(x), then ?
Therefore. the zeroes of the given fourth degree polynomial p(x) are
$<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

475 Views

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).

275 Views

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)

90 Views

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

126 Views

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)

102 Views

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

166 Views