%2F(3x%2B1).jpeg "(9x^3-18x^2-x+2)/(3x+1)")
Dividing Polynomials: (9x^3 - 18x^2 - x + 2) / (3x + 1)
This article will guide you through the process of dividing the polynomial (9x^3 - 18x^2 - x + 2) by the polynomial (3x + 1). We'll use the long division method for this.
Understanding Long Division with Polynomials
Long division with polynomials works similarly to long division with numbers. Here's a breakdown of the process:
- Set up: Write the dividend (9x^3 - 18x^2 - x + 2) inside the division symbol and the divisor (3x + 1) outside.
- Divide: Divide the leading term of the dividend (9x^3) by the leading term of the divisor (3x). This gives us 3x^2. Write this above the division symbol.
- Multiply: Multiply the divisor (3x + 1) by the term you just found (3x^2). This gives us 9x^3 + 3x^2. Write this below the dividend.
- Subtract: Subtract the result from the dividend. This will eliminate the leading term (9x^3).
- Bring Down: Bring down the next term from the dividend (-x).
- Repeat: Repeat steps 2-5 with the new polynomial.
- Continue: Keep repeating the process until you reach a remainder that has a degree less than the divisor.
Let's Do It!
3x^2 - 7x + 2
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3x+1 | 9x^3 - 18x^2 - x + 2
-(9x^3 + 3x^2)
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-21x^2 - x
-(-21x^2 - 7x)
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6x + 2
-(6x + 2)
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0
Explanation:
- Step 1: We set up the division problem as shown above.
- Step 2: We divide 9x^3 by 3x, getting 3x^2.
- Step 3: Multiply (3x + 1) by 3x^2 to get 9x^3 + 3x^2.
- Step 4: Subtract (9x^3 + 3x^2) from the dividend.
- Step 5: Bring down -x.
- Step 6: Repeat the process. We divide -21x^2 by 3x, getting -7x. Multiply (3x + 1) by -7x and subtract from the polynomial.
- Step 7: Bring down 2. Repeat the process. We divide 6x by 3x, getting 2. Multiply (3x + 1) by 2 and subtract.
We end up with a remainder of 0.
Conclusion
Therefore, (9x^3 - 18x^2 - x + 2) divided by (3x + 1) is 3x^2 - 7x + 2. This means we can rewrite the original expression as:
(9x^3 - 18x^2 - x + 2) = (3x + 1)(3x^2 - 7x + 2)