Find the reminder when (x^4-9x^2-2x+5) is divided by (x+3)
Reminder or remainder?
To find the remainder when the polynomial (x^4-9x^2-2x+5) is divided by (x+3), we can use polynomial long division. The dividend is (x^4-9x^2-2x+5) and the divisor is (x+3).
Step 1: Write the dividend (x^4-9x^2-2x+5) and the divisor (x+3) in long division format:
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x + 3 | x^4 - 9x^2 - 2x + 5
Step 2: Divide the highest degree term of the dividend (x^4) by the highest degree term of the divisor (x). The result is x^3.
Step 3: Multiply the divisor (x+3) by the quotient (x^3) and place the result (x^3(x+3)) below the dividend, aligning the terms:
x^3(x+3)
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x + 3 | x^4 - 9x^2 - 2x + 5
x^4 + 3x^3
Step 4: Subtract the result from step 3 from the dividend:
- 3x^3 - 2x + 5
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x + 3 | x^4 - 9x^2 - 2x + 5
x^4 + 3x^3
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- 12x^3 - 2x + 5
Step 5: Repeat steps 2-4 using the new dividend (-12x^3 - 2x + 5).
Step 2: Divide the highest degree term of the new dividend (-12x^3) by the highest degree term of the divisor (x). The result is -12x^2.
Step 3: Multiply the divisor (x+3) by the new quotient (-12x^2) and place the result (-12x^2(x+3)) below the new dividend, aligning the terms:
-12x^2(x+3)
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x + 3 | -12x^3 - 2x + 5
-12x^3 - 36x^2
Step 4: Subtract the result from step 3 from the new dividend:
34x^2 - 2x + 5
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x + 3 | -12x^3 - 2x + 5
-12x^3 - 36x^2
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34x^2 + 34x^2 + 5
Step 5: Repeat steps 2-4 using the new dividend (34x^2 + 34x^2 + 5).
Step 2: Divide the highest degree term of the new dividend (34x^2) by the highest degree term of the divisor (x). The result is 34x.
Step 3: Multiply the divisor (x+3) by the new quotient (34x) and place the result (34x(x+3)) below the new dividend, aligning the terms:
34x(x+3)
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x + 3 | 34x^2 + 34x^2 + 5
34x^2 + 102x
Step 4: Subtract the result from step 3 from the new dividend:
-99x + 5
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x + 3 | 34x^2 + 34x^2 + 5
34x^2 + 102x
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-99x + 5
Step 5: Repeat steps 2-4 using the new dividend (-99x + 5).
Step 2: Divide the highest degree term of the new dividend (-99x) by the highest degree term of the divisor (x). The result is -99.
Step 3: Multiply the divisor (x+3) by the new quotient (-99) and place the result (-99(x+3)) below the new dividend, aligning the terms:
-99(x+3)
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x + 3 | -99x + 5
-99x - 297
Step 4: Subtract the result from step 3 from the new dividend:
302
Step 5: Since the resulting dividend is a constant value (302), we have reached the end of the division process.
Therefore, the remainder when (x^4-9x^2-2x+5) is divided by (x+3) is 302.