Find the remainder when x⁴ + x³ + x² is divided by x-1
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[tex]\large\underline{\sf{Solution-}}[/tex]
Given polynomial is
[tex]\sf \: f(x) = {x}^{4} + {x}^{3} + {x}^{2} \\ \\ [/tex]
Now, we have to find the remainder when f(x) is divided by x - 1.
We know,
Remainder Theorem:-
This theorem states that if a polynomial f(x) of degree greater than or equals to one is divided by a linear polynomial x - a, then remainder is f(a).
So, using remainder theorem, remainder when f(x) is divided by x - 1 is
[tex]\sf \: Remainder = f(1) \\ \\ [/tex]
[tex]\sf \: = \: {(1)}^{4} + {(1)}^{3} + {(1)}^{2} \\ \\ [/tex]
[tex]\sf \: = \: 1 + 1 + 1 \\ \\ [/tex]
[tex]\sf \: = \: 3 \\ \\ [/tex]
Hence,
[tex]\bf\implies \:Remainder = 3 \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information
[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}[/tex]
Verified answer
Answer:
Find the remainder when x⁴ + x³ + x² is divided by x-1
Step-by-step explanation:
[tex]Given polynomial is \\ \\
\begin{gathered}\sf \: f(x) = {x}^{4} + {x}^{3} + {x}^{2} \\ \\ \\ \\ \end{gathered}f(x)=x4+x3+x2
Now, we have to find the remainder when f(x) is divided by x - 1.
We know,
Remainder Theorem:-
This theorem states that if a polynomial f(x) of degree greater than or equals to one is divided by a linear polynomial x - a, then remainder is f(a).
So, using remainder theorem, remainder when f(x) is divided by x - 1 is
\begin{gathered}\sf \: Remainder = f(1) \\ \\ \end{gathered}Remainder=f(1)
\begin{gathered}\sf \: = \: {(1)}^{4} + {(1)}^{3} + {(1)}^{2} \\ \\ \end{gathered}=(1)4+(1)3+(1)2
\begin{gathered}\sf \: = \: 1 + 1 + 1 \\ \\ \end{gathered}=1+1+1
\begin{gathered}\sf \: = \: 3 \\ \\ \end{gathered}=3
Hence,
\begin{gathered}\bf\implies \:Remainder = 3 \\ \\ \end{gathered}⟹Remainder=3
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