Find the quotient and remainder when F(x)=6x^3+4x^2-x+5 is divided by 3x^2+2x+1
To find the quotient and remainder when F(x) is divided by 3x^2+2x+1, we can use synthetic division.
Using synthetic division:
| 2 1 1 | | 6 4 -1 5 |
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6 16 31 | 5
The quotient is 6x + 16 and the remainder is 31.
Therefore, F(x) = (6x + 16)(3x^2 + 2x + 1) + 31.
To find the quotient and remainder when dividing a polynomial by another polynomial, we will use polynomial long division. Here's how:
Step 1: Arrange the dividend and the divisor in descending order of their degrees.
The dividend is F(x) = 6x^3 + 4x^2 - x + 5.
The divisor is D(x) = 3x^2 + 2x + 1.
Step 2: Divide the highest degree term of the dividend by the highest degree term of the divisor to obtain the first term of the quotient.
Dividing 6x^3 by 3x^2 gives us 2x.
Step 3: Multiply the entire divisor by the obtained term of the quotient.
2x * (3x^2 + 2x + 1) is equal to 6x^3 + 4x^2 + 2x.
Step 4: Subtract the product obtained in the previous step from the dividend.
(6x^3 + 4x^2 - x + 5) - (6x^3 + 4x^2 + 2x) is equal to -3x - x + 5.
Step 5: Repeat steps 2 to 4 with the new result obtained in step 4.
Dividing -3x by 3x^2 gives us -1/x.
Multiplying -1 by (3x^2 + 2x + 1) gives us -3x^2 - 2x - 1.
Subtracting the product from the previous result gives us (-3x - x + 5) - (-3x^2 - 2x - 1) which simplifies to -2x + 6.
Step 6: Repeat steps 2 to 4 with the new result obtained in step 5.
Dividing -2x by 3x^2 gives us -2/3x.
Multiplying -2/3 by (3x^2 + 2x + 1) gives us -2x - (4/3)x - 2/3.
Subtracting the product from the previous result gives us (-2x + 6) - (-2x - (4/3)x - 2/3) which simplifies to 6 + (10/3)x - 2/3.
At this point, we cannot divide further since the degree of the new result, 6 + (10/3)x - 2/3, is lower than the degree of the divisor.
Therefore, the quotient when F(x) is divided by 3x^2 + 2x + 1 is 2x - 1, and the remainder is 6 + (10/3)x - 2/3.
To find the quotient and remainder when F(x) is divided by 3x^2+2x+1, we can use polynomial long division.
Step 1: Write the divisor and dividend in descending powers of x:
Divisor: 3x^2+2x+1
Dividend: 6x^3+4x^2-x+5
Step 2: Start dividing the terms by the highest power of x in the divisor (3x^2).
3x^2 goes into 6x^3, 2x times.
- Multiply the divisor by 2x and subtract it from the dividend:
(2x)(3x^2+2x+1) = 6x^3 + 4x^2 + 2x
(6x^3+4x^2-x+5) - (6x^3+4x^2+2x) = -3x+5
Step 3: Bring down the next term (-3x):
Quotient so far: 2x
Dividend: -3x + 5
Step 4: Divide the new term by the highest power of x in the divisor (3x^2).
3x^2 goes into -3x, -x times.
- Multiply the divisor by -x and subtract it from the new dividend:
(-x)(3x^2+2x+1) = -3x^3 - 2x^2 - x
(-3x+5) - (-3x^3-2x^2-x) = 7x + 5
Step 5: Bring down the next term (7x):
Quotient so far: 2x - x
Dividend: 7x + 5
Step 6: Divide the new term by the highest power of x in the divisor (3x^2).
3x^2 goes into 7x, 2 times.
- Multiply the divisor by 2 and subtract it from the new dividend:
(2)(3x^2+2x+1) = 6x^2 + 4x + 2
(7x+5) - (6x^2+4x+2) = x + 3
Step 7: We have now reached the end, and we have the remainder x+3.
Quotient: 2x - x + 2
Remainder: x + 3
Therefore, the quotient when F(x) is divided by 3x^2+2x+1 is 2x - x + 2 and the remainder is x + 3.