-divided-by-(x%2B4).jpeg "(2x^3+7x^2-6x-8) Divided By (x+4)")
Dividing Polynomials: (2x^3 + 7x^2 - 6x - 8) ÷ (x + 4)
This article will walk through the process of dividing the polynomial 2x³ + 7x² - 6x - 8 by the binomial x + 4. We'll use the method of polynomial long division.
Step 1: Set up the division problem
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x + 4 | 2x³ + 7x² - 6x - 8
Step 2: Divide the leading terms
- Focus on the leading terms of the divisor (x) and the dividend (2x³).
- Ask: What do I multiply x by to get 2x³? The answer is 2x².
- Write 2x² above the division bar, aligned with the x² term.
2x²
x + 4 | 2x³ + 7x² - 6x - 8
Step 3: Multiply and subtract
- Multiply the entire divisor (x + 4) by 2x². This gives us 2x³ + 8x².
- Write this result below the dividend.
- Subtract the two expressions.
2x²
x + 4 | 2x³ + 7x² - 6x - 8
-(2x³ + 8x²)
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-x² - 6x
Step 4: Bring down the next term
- Bring down the next term of the dividend (-6x).
2x²
x + 4 | 2x³ + 7x² - 6x - 8
-(2x³ + 8x²)
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-x² - 6x
Step 5: Repeat steps 2-4
- Focus on the new leading term of the dividend (-x²) and the leading term of the divisor (x).
- Ask: What do I multiply x by to get -x²? The answer is -x.
- Write -x above the division bar, aligned with the x term.
2x² - x
x + 4 | 2x³ + 7x² - 6x - 8
-(2x³ + 8x²)
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-x² - 6x
-(-x² - 4x)
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-2x - 8
Step 6: Bring down the next term and repeat
- Bring down the next term of the dividend (-8).
- Focus on the new leading term of the dividend (-2x) and the leading term of the divisor (x).
- Ask: What do I multiply x by to get -2x? The answer is -2.
- Write -2 above the division bar, aligned with the constant term.
2x² - x - 2
x + 4 | 2x³ + 7x² - 6x - 8
-(2x³ + 8x²)
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-x² - 6x
-(-x² - 4x)
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-2x - 8
-(-2x - 8)
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0
Conclusion
The division is complete, and we have a remainder of 0. Therefore, the result of dividing 2x³ + 7x² - 6x - 8 by x + 4 is 2x² - x - 2.
In other words:
2x³ + 7x² - 6x - 8 = (x + 4)(2x² - x - 2)