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The polynomial \[P\left( x \right)={{x}^{4}}-2{{x}^{3}}+3{{x}^{2}}-ax+3a-7\] when divided by \[x+1\] leaves the remainder \[19\]. Find the value of \[a\]. Also find the remainder when \[P\left( x \right)\] is divided by \[x+2\].

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Last updated date: 14th Jul 2024
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Answer
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Hint: For solving this type of question we should know about the long division method. This method is used for dividing one large multi digit number into another large multi digit number. The formula for checking that our answer is correct or not is
Dividend = Divisor \[\times \] Quotient + Remainder

Complete step by step answer:
In this question it is asked to us that the polynomial \[P\left( x \right)={{x}^{4}}-2{{x}^{3}}+3{{x}^{2}}-ax+3a-7\] when divided by \[x+1\] leaves the remainder \[19\]. Find the value of \[a\]. Also, find the remainder when \[P\left( x \right)\] is divided by \[x+2\].
So, as we know, we can solve it directly by the remainder theorem.
The polynomial is given by \[P\left( x \right)\] -
\[P\left( x \right)={{x}^{4}}-2{{x}^{3}}+3{{x}^{2}}-ax+3a-7\]
Given that, the polynomial \[P\left( x \right)\] when divided by \[x+1\] leaves remainder \[19\]. Therefore, \[P\left( -1 \right)=19\] (By remainder theorem)
\[P\left( -1 \right)=19\Rightarrow {{\left( -1 \right)}^{4}}-2\times {{\left( -1 \right)}^{3}}+3{{\left( -1 \right)}^{2}}-\left( -1 \right)a+3a-7\]
So, if we solve this, then we find it as,
\[\begin{align}
  & \Rightarrow 1+2+3+a+3a-7=19 \\
 & \Rightarrow 6-7+4a=19 \\
 & \Rightarrow 4a-1=19 \\
 & \Rightarrow 4a=20 \\
 & \Rightarrow a=5 \\
\end{align}\]
So, here the value of \[a\] is equal to \[5\].
When put the value of \[a=5\], here,
Now, \[P\left( x \right)={{x}^{4}}-2{{x}^{3}}+3{{x}^{2}}-5x+3\times 5-7\]
\[\begin{align}
  & ={{x}^{4}}-2{{x}^{3}}+3{{x}^{2}}-5x+15-7 \\
 & ={{x}^{3}}-2{{x}^{3}}+3{{x}^{2}}-5x+8 \\
\end{align}\]
Now,
Remainder when the polynomial is divided by \[\left( x+2 \right)\] = \[P\left( -2 \right)\] (by remainder theorem)
\[\begin{align}
  & ={{\left( -2 \right)}^{4}}-2{{\left( -2 \right)}^{3}}+3{{\left( -2 \right)}^{2}}-5\left( -2 \right)+8 \\
 & =16+16+12+10+8 \\
 & =62 \\
\end{align}\]
Thus, the remainder of the polynomial \[P\left( x \right)\] when divided by \[\left( x+2 \right)\] is \[62\].

Note: During solving this question you should be careful to find the finding values of polynomials by direct form or by the remainder theorem. Because both will be different for different – different values of \[x\]. And you have to ensure that for which value that is going on.