%2F(x%2B4).jpeg "(2x^3+7x^2-6x-8)/(x+4)")
Dividing Polynomials: (2x^3 + 7x^2 - 6x - 8) / (x + 4)
This article will guide you through the process of dividing the polynomial (2x^3 + 7x^2 - 6x - 8) by the binomial (x + 4). We will use polynomial long division to accomplish this.
Understanding Polynomial Long Division
Polynomial long division follows a similar process to the traditional long division of numbers. Here's a breakdown:
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Set up the division: Write the dividend (2x^3 + 7x^2 - 6x - 8) inside the division symbol and the divisor (x + 4) outside.
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Divide the leading terms: Divide the leading term of the dividend (2x^3) by the leading term of the divisor (x). This gives us 2x^2. Write this result above the division symbol.
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Multiply the result by the divisor: Multiply 2x^2 by (x + 4), which results in 2x^3 + 8x^2. Write this below the dividend.
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Subtract: Subtract the result from the dividend. This leaves us with -x^2 - 6x - 8.
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Bring down the next term: Bring down the next term of the dividend (-6x) to create a new polynomial.
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Repeat steps 2-5: Divide the leading term of the new polynomial (-x^2) by the leading term of the divisor (x), which gives us -x. Write this above the division symbol.
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Multiply and subtract: Multiply -x by (x + 4) and subtract the result from the current polynomial.
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Continue: Repeat steps 6-7 until you reach a remainder with a degree lower than the divisor.
Performing the Division
Let's apply the steps to our problem:
2x^2 - x - 2
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x + 4 | 2x^3 + 7x^2 - 6x - 8
-(2x^3 + 8x^2)
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-x^2 - 6x
-(-x^2 - 4x)
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-2x - 8
-(-2x - 8)
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0
The Result
After performing polynomial long division, we find that:
(2x^3 + 7x^2 - 6x - 8) / (x + 4) = 2x^2 - x - 2
Therefore, (2x^3 + 7x^2 - 6x - 8) is divisible by (x + 4) with a quotient of 2x^2 - x - 2 and a remainder of 0.