Math, asked by peeyushkumar266, 5 hours ago

Please do the test correctly and fast please i need it urvently please...​

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Answered by Anushkas7040
2

Step-by-step explanation:

1)

For 0 of (x-1)

=x-1=0

=x=1

Therefore,

p(1) = {1}^{2}  - 2(1) + 1  \:  \:  \:  \:  \:  \:  \:  \: (by \: remainder \: theorem) \\  =  > 1 - 2 + 1 \\  =  > 0

2)

(x - 1) - ( {x}^{2}  - 1) \\  =  > x - 1 -  {x}^{2}  - 1 \\  =  >  -  {x}^{2}  + x - 2

Therefore, the coefficient of x² is -1

3)

(x -  \frac{1}{x} )(x +  \frac{1}{x} )( {x}^{2} +  \frac{1}{ {x}^{2} }  )  \:  \:  \:  \:  \:  \:  \: \\ using \: identity \: (a + b)(a - b) =  {a}^{2} -  {b}^{2}  \\  \\  =  &gt; ( {x}^{2}  -  \frac{1}{ {x}^{2} })(  {x}^{2} +  \frac{1}{ {x}^{2} } )  \\  \\  =  &gt; ( {x}^{4}  -  \frac{1}{ {x}^{4} })</p><p>

4)

p(x) =  {x}^{19}   +  1 \\ g(x) = x + 1

For zero of g(x)

x+1=0

=x=-1

Therefore,

p( - 1) =  {( - 1)}^{19}  + 1 \:  \:  \:  \:  \: (by \: remainder \: theorem) \\ =  &gt;  - 1 + 1 = 0 \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \: </p><p>

Since g(x) is exactly divisible by p(x)

Therefore g(x) is a factor of p(x)

[By factor theorem]

5)

a)

b)

125 {n}^{3}   + 27 {y}^{3}  \\  =  &gt;  {(5n)}^{3}  +  {(3y)}^{2}  \\  =  &gt; (5n + 3y)( {(5n)}^{2}   - (5n)(3y) +  {(3y)}^{2} ) \\  =  &gt; (5n + 3y)(25 {n}^{2}  - 15ny + 9 {y}^{2} ) \\ </p><p></p><p>

c)

 {(x  -  y)}^{3}  +  {(y - z)}^{3}  + (z - x)^{3}  \\  =  &gt; since(x - y) + (y - z) + (z - x) = 0 \\ therefore \:  {(x  -  y)}^{3}  +  {(y - z)}^{3}  + (z - x)^{3}  = 3(x - y)(y - z)(z - x) \\  =  &gt; 3(xy - xz -  {y}^{2}  + yz)(z - x) \\  =  &gt; (3)(xyz  +  {x}^{2}y  - x {z}^{2}  +  {x}^{2} z -  {y}^{2} z +  {y}^{2} x + y {z}^{2}  - xyz) \\  =  &gt; 3(  - {x}^{2}y +  {x}^{2}z   -  x {z}^{2}  -  {y}^{2}z +  {y}^{2} x + y {z}^{2} ) \\   =  &gt; 3(  - {x}^{2} y +  {x}^{2} z -  {y}^{2} x +  {y}^{2} z - x {z}^{2}  + y {z}^{2} ) \\  =  &gt; 3(  - {x}^{2} (y  -  z) -  {y}^{2} (x - z) -  {z}^{2} (x - y)) \\  =  &gt; 3( -  {x}^{2}  -  {y}^{2}  -  {z}^{2} )(y - z)(x - z)(x - y)

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