Use long division to find the remainder when f(x)=3x^5-7x^3-11x^2+2 is divided by x-2.
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polynomial long division online
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Long division of polynomials - calculator - Wolfram Library Archive
When page be open in rectangle Divide type 3x^5-7x^3-11x^2
In rectangle by type x-2
Cliclk Do it and you will see result step-by-step
Remainder = - 4
To find the remainder when f(x) = 3x^5 - 7x^3 - 11x^2 + 2 is divided by x - 2, we will use long division.
Step 1: Write the polynomial in descending order of powers of x:
3x^5 - 7x^3 - 11x^2 + 2
Step 2: Divide the highest power of x in the polynomial (3x^5) by the highest power of x in the divisor (x):
3x^5 / x = 3x^4
Step 3: Multiply 3x^4 by the divisor (x - 2):
3x^4 * (x - 2) = 3x^5 - 6x^4
Step 4: Subtract the result from step 3 from the original polynomial:
(3x^5 - 7x^3 - 11x^2 + 2) - (3x^5 - 6x^4) = -7x^3 - 11x^2 + 6x^4 + 2
Step 5: Repeat steps 2-4 with the remaining terms of the polynomial:
(-7x^3) / x = -7x^2
-7x^2 * (x - 2) = -7x^3 + 14x^2
(-7x^3 - 11x^2 + 6x^4 + 2) - (-7x^3 + 14x^2) = -11x^2 - 6x^4 + 2 - (-14x^2)
Step 6: Divide again:
(-11x^2) / x = -11x
-11x * (x - 2) = -11x^2 + 22x
(-11x^2 - 6x^4 + 2 - (-14x^2)) - (-11x^2 + 22x) = -6x^4 - 14x^2 + 22x + 2
Step 7: Continue dividing until we reach a term of lower degree than the divisor (x - 2):
(-6x^4) / x = -6x^3
-6x^3 * (x - 2) = -6x^4 + 12x^3
(-6x^4 - 14x^2 + 22x + 2) - (-6x^4 + 12x^3) = -14x^2 + 22x + 2 + 12x^3
Step 8: Divide again:
(-14x^2) / x = -14
-14 * (x - 2) = -14x + 28
(-14x^2 + 22x + 2 + 12x^3) - (-14x + 28) = 22x + 12x^3 + 2
Step 9: Divide one more time:
(22x) / x = 22
22 * (x - 2) = 22x - 44
(22x + 12x^3 + 2) - (22x - 44) = 12x^3 + 2 + 44
Since the degree of 12x^3 is less than x, our division is complete. The remainder of f(x) divided by (x - 2) is 12x^3 + 2 + 44, or simply, 12x^3 + 46.
Sure! To find the remainder when f(x) is divided by x-2 using long division, we can follow these steps:
1. Write the dividend (f(x)) and the divisor (x-2) in the long division format:
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x - 2 | 3x^5 - 7x^3 - 11x^2 + 2
2. Start by dividing the first term of the dividend (3x^5) by the first term of the divisor (x), which gives us 3x^4.
3x^4
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x - 2 | 3x^5 - 7x^3 - 11x^2 + 2
3. Multiply the divisor (x - 2) by the quotient obtained in the previous step (3x^4) and subtract it from the first term of the dividend. Bring down the rest of the terms.
3x^4
___________________
x - 2 | 3x^5 - 7x^3 - 11x^2 + 2
- (3x^5 - 6x^4)
_________________
- x^4 - 11x^2 + 2
4. Repeat steps 2 and 3 with the updated dividend (-x^4 - 11x^2 + 2).
5. Divide the first term of the updated dividend (-x^4) by the first term of the divisor (x), which gives you -x^3.
6. Multiply the divisor (x - 2) by the quotient obtained in the previous step (-x^3) and subtract it from the updated dividend. Bring down the rest of the terms.
7. Repeat steps 5 and 6 with the updated dividend until all terms have been divided.
8. Once you reach the end of the division, the term left in the dividend is the remainder.
In this case, the remainder is: - x^2 + 2.