
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 \[x+2\].
Answer
410.1k+ views
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.
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.
Recently Updated Pages
Questions & Answers - Ask your doubts

Master Class 11 Accountancy: Engaging Questions & Answers for Success

Master Class 11 Science: Engaging Questions & Answers for Success

Master Class 12 Economics: Engaging Questions & Answers for Success

Master Class 12 Maths: Engaging Questions & Answers for Success

Master Class 12 Biology: Engaging Questions & Answers for Success

Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

Difference Between Plant Cell and Animal Cell

What is pollution? How many types of pollution? Define it

What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.

Distinguish between Conventional and nonconventional class 9 social science CBSE
