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Class 10 Class 12

Polynomials

Find the zeroes of the quadratic polynomial x² + 5x + 6 and verify the relationship between the zeroes and the coefficients.

We have,
$straight f left parenthesis straight x right parenthesis space equals space straight x squared plus 5 straight x plus 6$
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$rightwards double arrow$ $straight f left parenthesis straight x right parenthesis space equals space straight x left parenthesis straight x plus 3 right parenthesis plus 2 left parenthesis straight x plus 3 right parenthesis$
$rightwards double arrow$ $straight f left parenthesis straight x right parenthesis space equals space left parenthesis straight x plus 2 right parenthesis left parenthesis straight x plus 3 right parenthesis$
For the zeroes of the polynomial
$space straight f left parenthesis straight x right parenthesis space equals space 0$
$rightwards double arrow$ (x + 2) (x + 3) = 0
$rightwards double arrow$ x + 2 = 0 or x + 3 = 0
$rightwards double arrow$ x = -2 or x = -3
Thus, the zeroes of f(x ) =x² + 5x + 6 are - α= –2 and β = –3
Now,
Sum of the zeroes = $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
= $space space minus fraction numerator coefficient space of space straight x over denominator coefficient space of space straight x squared end fraction$

Product of the zeroes = $straight alpha. straight beta space equals space left parenthesis negative 2 right parenthesis cross times left parenthesis negative 3 right parenthesis space equals space 6$
= $fraction numerator constant space term over denominator coefficient space of space straight x squared end fraction.$

124 Views

Find a quadratic polynomial whose zeroes are 1 and (-3). Verify the relation between the coefficients and zeroes of the polynomial.

Let the quadratic polynomial be ax² + bx + c, and its zeroes be α and β. Then
α = 1 and β = - 3
∴ Sum of the zeroes (α + β) = 1 + (– 3) = – 2 and, product of zeroes (a+ p) = 1 x (–3) = –3
Hence, the quadratic polynomial
= x² – (α + β) x + αβ
= x² – (– 2)x + (– 3)
= x² + 2x – 3
Now, sum of the zeroes

equals 1 plus left parenthesis negative 3 right parenthesis equals fraction numerator negative 2 over denominator 1 end fraction equals fraction numerator Coefficient space of space straight x over denominator Coefficient space of space straight x squared end fraction

and product of the zeroes

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

equals fraction numerator Constant space term over denominator Coefficient space of space straight x squared end fraction

2920 Views

Kind 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.

Let the quadratic polynomial be ax³ +bx² +cx + d and its zeroes be α, β and ϒ.
We have,

straight alpha plus straight beta plus straight gamma space equals space 2 over 1 equals negative fraction numerator Coefficient space of space straight x squared over denominator Coefficient space of space straight x cubed end fraction equals negative straight b over straight a

αβ plus βγ plus γα equals fraction numerator negative 7 over denominator 1 end fraction equals fraction numerator Coefficient space of space straight x over denominator Coefficient space of space straight x cubed end fraction equals straight c over straight a

and

space space space space αβγ equals fraction numerator negative 14 over denominator 1 end fraction equals negative fraction numerator Constant space term over denominator Coefficient space of space straight x cubed end fraction equals fraction numerator negative straight d over denominator straight a end fraction

When we put a = 1, then b = -2, c=-7, d = 14.
Therefore, the required cubic polynomial be x³ – 2x² – 7x + 14.

270 Views

Kind 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 –9, –11, 30 respectively.

Let the quadratic polynomial be ax³ – bx² + cx + d and its zeroes be α, β and y.
We have,

straight alpha plus straight beta plus straight gamma space equals space fraction numerator negative 9 over denominator 1 end fraction space equals space minus fraction numerator Coefficient space of space straight x squared over denominator Coefficient space of space straight x cubed end fraction equals negative straight b over straight a

αβ plus βγ plus γα equals fraction numerator negative 11 over denominator 1 end fraction equals fraction numerator Coefficient space of space straight x over denominator Coefficient space of space straight x cubed end fraction equals straight c over straight a

and

αβγ space equals space 30 over 1 space equals space minus fraction numerator Constant space term over denominator Coefficient space of space straight x cubed end fraction space equals space fraction numerator negative straight d over denominator straight a end fraction

When we put a = 1, then b = 9, c = -11, d = 30.
Therefore, the requried cubic polynomial is

straight x cubed plus 9 straight x squared minus 11 straight x minus 30.

114 Views

Find the zeroes of the quadratic polynomial 4x² – 4x – 3 and verify the relation between the zeroes and its coefficients.

4x² – 4x – 3
= 4x² – 6x + 2x – 3
= 2x (2x – 3) + 1 (2x – 3)
= (2x – 3) (2x + 1)
So, 4x² – 4x – 3 = 0 ⇒ (2x – 3) (2x + 1) = 0
⇒ 2x – 3 = 0 or 2x + 1 = 0
⇒ x = 3/2 or x = – 1/2
∴ Zeroes of $4 straight x squared minus 4 straight x minus 3$ are $3 over 2 space and space minus 1 half$
Now,
Sum of zeroes = $3 over 2 plus open parentheses negative 1 half close parentheses space equals space 1 equals fraction numerator negative left parenthesis negative 4 right parenthesis over denominator 4 end fraction equals negative straight b over straight a$

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and product of zeroes = $3 over 2 cross times open parentheses negative 1 half close parentheses equals fraction numerator negative 3 over denominator 4 end fraction equals straight c over straight a$

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