P 1 ( x ) = 3 x 2 + 10 x + 8 P 2 ( x ) = x 3 + x 2 + 2 x + t are two polynomials. When one of the factors of P 1 ( x ) divides P 2 ( x ) , 2 is the remainder obtained. That factor is also a factor of the polynomial P 3 ( x ) = 2 ( x + 2 ) Find the value of ‘t’.

Question

P     1      (   x   )   =   3        x     2      +   10   x   +   8           P     2      (   x   )   =        x     3      +        x     2      +   2   x   +   t   are two polynomials.  When one of the factors of         P     1      (   x   )  divides         P     2      (   x   ) , 2 is the remainder obtained.  That factor is also a factor of the polynomial         P     3      (   x   )   =   2   (   x   +   2   )   Find the value of ‘t’.

Haleigh Russo · Accepted Answer

Let the factor be (x-a).  Using factor theorem,         P     3      (   a   )   =   0           P     3      (   x   )   =   2   (   x   +   2   )           P     3      (   a   )   =   2   (   a   +   2   )   =   0      &amp;#x21D2;   a   =   &amp;#x2212;   2   So, the factor is    (   x   +   2   ) . Verifying that this is also a factor of         P     1      (   x   )  using factor theorem,          P     1      (   &amp;#x2212;   2   )   =   3   &amp;#x00D7;   (   &amp;#x2212;   2        )     2      +   10   &amp;#x00D7;   (   &amp;#x2212;   2   )   +   8           P     1      (   &amp;#x2212;   2   )   =   3   &amp;#x00D7;   4   &amp;#x2212;   20   +   8   =   0   By remainder theorem, when         P     2      (   x   )  is divided by (x+2), the remainder is          P     2      (   &amp;#x2212;   2   )           P     2      (   &amp;#x2212;   2   )   =   (   &amp;#x2212;   2        )     3      +   (   &amp;#x2212;   2        )     2      +   2   &amp;#x00D7;   (   &amp;#x2212;   2   )   +   t           P     2      (   &amp;#x2212;   2   )   =   &amp;#x2212;   8   +   4   &amp;#x2212;   4   +   t   =   t   &amp;#x2212;   8   Given remainder is 2. So,          P     2      (   &amp;#x2212;   2   )   =   2      &amp;#x21D2;   t   &amp;#x2212;   8   =   2      &amp;#x21D2;   t   =   10

P 1 ( x ) = 3 x 2 + 10 x + 8 P...

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