Question:
The remainder when the polynomial
\[
2x^5-3x^4+5x^3-3x^2+7x-9
\]
is divided by
\[
x^2-x-3
\]
is
The remainder when the polynomial
\[
2x^5-3x^4+5x^3-3x^2+7x-9
\]
is divided by
\[
x^2-x-3
\]
is
Show Hint
When dividing by a quadratic polynomial, reduce higher powers using the relation obtained from the divisor. This avoids long division and quickly yields the remainder.
Updated On: Jun 26, 2026
- \(-41x-3\)
- \(41x+3\)
- \(41x-3\)
- \(-41x+3\)
Show Solution
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The Correct Option is B
Solution and Explanation
Step 1: Let the remainder be \(R(x)=ax+b\).
Since the divisor is a quadratic polynomial, \[ x^2-x-3, \] the remainder must be of degree less than \(2\). Therefore, \[ R(x)=ax+b. \]
Step 2: Use the relation from the divisor.
Since \[ x^2-x-3=0, \] we get \[ x^2=x+3. \] Using this repeatedly, \[ x^3=x(x+3)=x^2+3x=(x+3)+3x=4x+3. \] \[ x^4=x(4x+3)=4x^2+3x =4(x+3)+3x =7x+12. \] \[ x^5=x(7x+12) =7x^2+12x =7(x+3)+12x =19x+21. \]
Step 3: Substitute these values into the polynomial.
\[ P(x)=2x^5-3x^4+5x^3-3x^2+7x-9 \] \[ =2(19x+21)-3(7x+12)+5(4x+3)-3(x+3)+7x-9 \] \[ =38x+42-21x-36+20x+15-3x-9+7x-9 \] Collecting \(x\)-terms, \[ (38-21+20-3+7)x =41x \] Collecting constants, \[ 42-36+15-9-9 =3 \] Hence, \[ R(x)=41x+3. \]
Step 4: Final conclusion.
Therefore, the remainder is \[ \boxed{41x+3} \]
Since the divisor is a quadratic polynomial, \[ x^2-x-3, \] the remainder must be of degree less than \(2\). Therefore, \[ R(x)=ax+b. \]
Step 2: Use the relation from the divisor.
Since \[ x^2-x-3=0, \] we get \[ x^2=x+3. \] Using this repeatedly, \[ x^3=x(x+3)=x^2+3x=(x+3)+3x=4x+3. \] \[ x^4=x(4x+3)=4x^2+3x =4(x+3)+3x =7x+12. \] \[ x^5=x(7x+12) =7x^2+12x =7(x+3)+12x =19x+21. \]
Step 3: Substitute these values into the polynomial.
\[ P(x)=2x^5-3x^4+5x^3-3x^2+7x-9 \] \[ =2(19x+21)-3(7x+12)+5(4x+3)-3(x+3)+7x-9 \] \[ =38x+42-21x-36+20x+15-3x-9+7x-9 \] Collecting \(x\)-terms, \[ (38-21+20-3+7)x =41x \] Collecting constants, \[ 42-36+15-9-9 =3 \] Hence, \[ R(x)=41x+3. \]
Step 4: Final conclusion.
Therefore, the remainder is \[ \boxed{41x+3} \]
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