Difference between revisions of "Remainder Theorem"
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== Example == | == Example == | ||
Find the remainder when the polynomial <math>p(x)=x^ | Find the remainder when the polynomial <math>p(x)=3x^3+x^2+2x+5</math> is divided by <math>x+1</math>. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
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| colspan="7" style="border-bottom: solid 5px blue" |<math>3x^2 -2x+4</math> | | colspan="7" style="border-bottom: solid 5px blue" |<math>3x^2 -2x+4</math> | ||
|- | |- | ||
| rowspan="7" style="border-right: solid 5px blue ;vertical-align:top" |<math>x+1</math> | | rowspan="7" style="border-right: solid 5px blue ;vertical-align:top" |'''<math>x+1</math>''' | ||
|<math>3x^3</math> | |<math>3x^3</math> | ||
|<math>+</math> | |<math>+</math> | ||
Line 23: | Line 23: | ||
|<math>+</math> | |<math>+</math> | ||
|<math>3x^2</math> | |<math>3x^2</math> | ||
| | | colspan="4" | | ||
| | |||
|- | |- | ||
| rowspan="5" | | | rowspan="5" | | ||
Line 40: | Line 37: | ||
|<math>-</math> | |<math>-</math> | ||
|<math>2x</math> | |<math>2x</math> | ||
| | | colspan="2" | | ||
| | |||
|- | |- | ||
| rowspan="3" | | | rowspan="3" | | ||
Line 56: | Line 52: | ||
|- | |- | ||
| colspan="3" | | | colspan="3" | | ||
|1 | |'''1''' | ||
|} | |} | ||
Here, quotient = <math>3x^2 -2x+4</math> | |||
Remainder = <math>1</math> | |||
'''Verification :''' | |||
Given, the divisor is <math>x+1</math>, i.e. it is a factor of the given polynomial <math>p(x)</math> | |||
Let <math>x+1=0</math> | |||
<math>x=-1</math> | |||
Substituting <math>x=-1</math> in <math>p(x)</math>, | |||
<math>p(x)=3x^3+x^2+2x+5</math> | |||
<math>p(-1-)=3(-1)^3+(-1)^2+2(-1)+5</math> | |||
<math>p(-1-)=3(-1)+1-2+5</math> | |||
<math>p(-1-)=-3+1-2+5</math> | |||
<math>p(-1-)=1</math> | |||
Remainder = Value of <math>p(x)</math> at <math>x=-1</math>. | |||
Hence proved the remainder theorem. |
Latest revision as of 11:26, 10 May 2024
The Remainder theorem formula is used to find the remainder when a polynomial is divided by a linear polynomial.
Remainder Theorem
The Remainder theorem states that "when a polynomial is divided by a linear polynomial , then the remainder is "
Example
Find the remainder when the polynomial is divided by .
1 |
Here, quotient =
Remainder =
Verification :
Given, the divisor is , i.e. it is a factor of the given polynomial
Let
Substituting in ,
Remainder = Value of at .
Hence proved the remainder theorem.