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Class 10 Class 12

Polynomials

Verify that the numbers given along side of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficient in each case.
x³ – 4x² + 5x – 2, 2, 1, 1.

The general form of cubic polynomial is ax³ + bx² + cx + d when, we compare the given cubic polynomial x³ - 4x² + 5x – 2 with general form, we get
a = 1, b = -4
c = 5 and d = -2

Let space space space space space space space space space space space space space space space space space space space space space space straight P left parenthesis straight x right parenthesis space equals space ax cubed plus bx squared plus cx plus straight d space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight P left parenthesis 2 right parenthesis space equals space 1 left parenthesis 2 right parenthesis cubed plus left parenthesis negative 4 right parenthesis space left parenthesis 2 right parenthesis squared plus space 5 space left parenthesis 2 right parenthesis space plus space left parenthesis negative 2 right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 8 minus 16 plus 10 minus 2 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals negative 8 plus 8 equals 0

straight P left parenthesis 1 right parenthesis space equals space 1 left parenthesis 1 right parenthesis cubed plus left parenthesis negative 4 right parenthesis left parenthesis 1 right parenthesis squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space plus space left parenthesis 5 right parenthesis space left parenthesis 1 right parenthesis space plus space left parenthesis negative 2 right parenthesis space space space space space space space space space equals space 1 minus 4 plus 5 minus 2 space space space space space space space space space equals space 6 minus 6 space equals space 0

and

straight P left parenthesis 1 right parenthesis space equals space 1 left parenthesis 1 right parenthesis cubed plus left parenthesis negative 4 right parenthesis left parenthesis 1 right parenthesis squared plus left parenthesis 5 right parenthesis space left parenthesis 1 right parenthesis space plus space left parenthesis negative 2 right parenthesis

space equals 1 minus 4 plus 5 minus 2 space space equals space 6 minus 6 space equals space 0

Verification of relationship between zeroes and coefficients.
Let

space space space space straight alpha space equals space 2 comma space space space space straight beta equals 1 comma space space straight gamma equals 1

Now,

<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>

<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>

equals negative fraction numerator Coefficient space of space straight x squared over denominator Coefficient space of space straight x cubed end fraction

αβ plus βγ plus γα space equals space left parenthesis 2 right parenthesis left parenthesis 1 right parenthesis plus left parenthesis 1 right parenthesis left parenthesis 1 right parenthesis plus left parenthesis 1 right parenthesis left parenthesis 2 right parenthesis

equals 2 plus 1 plus 2 equals 5 over 1

equals fraction numerator Coefficient space of space straight x over denominator Coefficient space of space straight x cubed end fraction

αβγ space equals space left parenthesis 2 right parenthesis space left parenthesis 1 right parenthesis space left parenthesis 1 right parenthesis space equals space 2

equals fraction numerator negative left parenthesis negative 2 right parenthesis over denominator 1 end fraction

equals negative fraction numerator Constant space term over denominator Coefficient space of space straight x cubed end fraction.

132 Views

Find the zeroes of the following polynomial and verify the relationship between the zeroes and the coefficients

5 square root of 5 straight y squared plus 20 straight y plus 3 square root of 5

We have,
 $5 square root of 5 straight y squared plus 20 straight y plus 3 square root of 5$
 $equals space 5 square root of 5 straight y squared plus 15 straight y plus 5 straight y plus 3 square root of 5 equals space space space space open parentheses 5 straight y plus square root of 5 close parentheses space open parentheses square root of 5 straight y plus 3 close parentheses$
So, the value of $5 square root of 5 straight y squared plus 20 straight y plus 3 square root of 5$ is zero when $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
i.e., when $space space space space space space straight y equals fraction numerator negative square root of 5 over denominator 5 end fraction space space or space space straight y equals fraction numerator negative 3 over denominator square root of 5 end fraction$
Therefore, the zeroes of $5 square root of 5 straight y squared plus 20 straight y plus 3 square root of 5$ are $fraction numerator negative square root of 5 over denominator 5 end fraction space and space fraction numerator negative 3 over denominator square root of 5 end fraction$
Sum of zeroes = $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
 $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

Product of zeroes = $space space open parentheses fraction numerator negative square root of 5 over denominator 5 end fraction close parentheses cross times fraction numerator negative 3 over denominator square root of 5 end fraction equals fraction numerator 3 square root of 5 over denominator 5 square root of 5 end fraction$
 $equals fraction numerator Constant space term over denominator Coefficient space of space straight y squared end fraction.$

90 Views

Verify that 3, -1, $fraction numerator negative 1 over denominator 3 end fraction$ are the zeroes of the cubic polynomial p(x) = 3x³ – 5x² –11x – 3 then verify the relationship between the zeroes and the coefficients.

Since, most general form of cubic polynomial is ax³ + bx² + cx + d When, we compare the given cubic polynomial with the general form, we get
a = 3, b = –5
c = – 11 and d = – 3
Let P(x) ax^3, + bx² + cx + d
Now, P(3) = 3 (3)³ + (– 5) (3)²
+ (-11) (3) + (-3)
= 81 - 45 - 33 - 3
= 0

space space straight P left parenthesis negative 1 right parenthesis space equals space 3 left parenthesis negative 1 right parenthesis cubed plus left parenthesis negative 5 right parenthesis left parenthesis negative 1 right parenthesis squared space space space space space space space space space space space space space space space plus left parenthesis negative 11 right parenthesis space left parenthesis negative 1 right parenthesis space plus space left parenthesis negative 3 right parenthesis

space equals 3 left parenthesis negative 1 right parenthesis minus 5 plus 11 minus 3 space equals negative 3 minus 5 plus 11 minus 3 space equals negative 11 plus 11 equals 0

and

straight P open parentheses negative 1 third close parentheses space equals space 3 open parentheses negative 1 third close parentheses cubed plus left parenthesis 5 right parenthesis open parentheses negative 1 third close parentheses squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space

plus left parenthesis negative 11 right parenthesis open parentheses negative 1 third close parentheses plus left parenthesis negative 3 right parenthesis

equals 3 cross times fraction numerator negative 1 over denominator 27 end fraction minus 5 cross times 1 over 9 plus 11 over 3 minus 3 over 1

equals negative 1 over 9 minus 5 over 9 plus 11 over 3 minus 3 over 1

equals fraction numerator negative 1 minus 5 plus 33 minus 27 over denominator 9 end fraction

<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>

Therefore, the given values 3, -1 and

fraction numerator negative 1 over denominator 3 end fraction

are the zeroes of 3x³ – 5x² – 11x – 3.
Verification of relationship between zeroes and the coefficients the zeroes of 3x³ – 5x² – 11x – 3.
Since, most general form of cubic polynomial is ax3 + bx2 + cx + d

Since, most general form of cubic polynomial is ax3 + bx2 + cx + d

equals 1 equals fraction numerator negative left parenthesis negative 3 right parenthesis over denominator 3 end fraction

equals negative fraction numerator Constant space term over denominator Coefficient space of space straight x cubed end fraction.

2259 Views

Find the zeroes of x² –2 and verify the relationship between the zeroes and the coefficients.

We have

<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>

So, the value of

straight x squared minus 2

is zero when

left parenthesis straight x plus square root of 2 right parenthesis equals 0 space space space space or comma space space space straight x minus square root of 2 equals 0.

i.e. when

straight x equals negative square root of 2 space space space space or space straight x equals square root of 2

Therefore, the zeroes of

straight x squared minus 2 space are space minus square root of 2 space and space square root of 2.

Now,
Sum of zeroes

equals open parentheses negative square root of 2 close parentheses plus left parenthesis square root of 2 right parenthesis space equals space 0 over 1 equals fraction numerator Coefficient space of space straight x 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>

280 Views

Find the zeroes of a quadratic polynomial 5x² – 4 – 8x and verify the relationship between the zeroes and the coefficients.

We have, 5x² – 8x – 4
= 5x² – 10x + 2x – 4
= 5x (x–2) + 2 (x – 2)
= (5x + 2) (x – 2)
So, the value of 5x² – 8x – 4 is zero when 5x + 2 = 0 or x – 2 = 0
i.e., when x = –2/5 or x = 2
Therefore, the zeroes of 5x² – 8x – 4 are –2/5 and 2.
Sum of zeroes = $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

= $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

Product of zeroes = $negative 2 over 5 cross times 2 equals fraction numerator negative 4 over denominator 5 end fraction$

= $fraction numerator Constant space term over denominator Coefficient space of space straight x squared end fraction.$