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    Polynomials
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    Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
          4 straight s squared minus 4 straight s plus 1
         
    We have,
    4s2 – 4s + 1 = 4s2 – 2s –2s + 1 = (2s–1)(2s– 1)
    So, the value of 4s2 – 4s + 1 is zero, when 2s – 1 = 0 or 2s – 1 = 0
    i.e.  when   straight s equals 1 half space or space straight s equals 1 half
    Therefore, the zeroes of 4s2 – 4s + 1 are 1 half space and space 1 half. Now,
         Sum of zeroes =  1 half plus 1 half

                                 equals 1 over 1 equals fraction numerator negative left parenthesis Coefficient space of space straight s right parenthesis over denominator Coefficient space of space straight s squared end fraction

    Product of zeroes  = 1 half cross times 1 half equals 1 fourth
                                  <pre>uncaught exception: <b>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</b><br /><br />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<br />#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>
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    Answer
    Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
    6 straight x squared minus 3 minus 7 straight x
          
         
    We have,6x2–7x–3 = 6x2– 9x + 2x – 3
    = 3x(2x – 3) + 1(2x – 3)
    = (3x + 1) (2x – 3)
    So, the value of 6x2 – 7 – 3 is zero when (3a + 1) = 0 or (2x – 3) = 0
    i.e.                 <pre>uncaught exception: <b>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</b><br /><br />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<br />#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>
    Now, 
       Sum of zeroes  = <pre>uncaught exception: <b>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</b><br /><br />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<br />#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>

                                space space space equals fraction numerator negative left parenthesis Coefficient space of space straight x right parenthesis over denominator Coefficient space of space straight x squared end fraction

    Product of zeroes = <pre>uncaught exception: <b>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</b><br /><br />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<br />#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>

                                = fraction numerator Constant space term over denominator Coefficient space of space straight x squared end fraction.
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    Answer
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    Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
    4 straight u squared plus 8 straight u
          
         

    We have,
    4u2 + 8u = 4u (u + 2)
    So, the value of 4u2 + 8u is zero when 4u = 0 or u + 2 = 0
    i.e., when u = 0 or u = –2
    Therefore, the zeroes of 4u2 + 8u are 0 and - 2
    Now,
    Sum of zeroes = 0 + (-2) = fraction numerator negative 2 over denominator 1 end fraction
                                             equals fraction numerator negative left parenthesis Coefficient space of space straight u right parenthesis over denominator left parenthesis Coefficient space of space straight u squared right parenthesis end fraction
    Product of zeroes = 0(-2) = 0 over 1
                                = fraction numerator Constant space term over denominator Coefficient space of space straight u squared end fraction

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    Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
         straight x squared minus 2 straight x minus 8

    We have,
          space space space straight x squared minus 2 straight x minus 8 space equals space straight x squared minus 4 straight x plus 2 straight x minus 8
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    So, the value of x2 – 2x - 8 is zero when x - 4 = 0 or x + 2 = 0.
    i.e., when x = 4 or x = –2.
    Therefore, the zeroes of x2 – 2x – 8 are 4 and - 2. Now,
       Sum of zeroes = (4) + (-2)
                               equals 2 over 1 equals fraction numerator negative left parenthesis Coefficient space of space straight x right parenthesis over denominator Coefficient space of space straight x squared end fraction
    Product of zeroes = (4) (-2)
                                 equals fraction numerator negative 8 over denominator 1 end fraction equals fraction numerator Constant space term over denominator Coefficient space of space straight x squared end fraction.


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    The graph of y = p(x) are given in the figure below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.
    (i) Graph y = p(x) does not cut the x-axis at any point, so the given polynomial has no zero. (ii) Graph y = p(x) cuts the x-axis at one point, so the given polynomial has one zero. (iii) Graph y = p(x) cuts the x-axis at three points, so the given polynomial has three zeroes. (iv) Graph y = p(x) cuts the x-axis at two points, so the given polynomial has two zeroes. (v) Graph y = p(x) cuts the x-axis at four points, so the given polynomial has four zeroes. (vi) Graph y = p(x) cuts the x-axis at three points, so the given polynomial has three zeroes.
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