-div(x-2)%3D%24.jpeg "$(x^3-3x^2-7x+6) Div(x-2)=$")
Dividing Polynomials: $(x^3-3x^2-7x+6) \div (x-2)$
This article will explore the process of dividing the polynomial $(x^3-3x^2-7x+6)$ by $(x-2)$. We will use polynomial long division to achieve this.
Polynomial Long Division
Polynomial long division is similar to the long division of numbers. Here are the steps involved:
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Set up the division problem. Write the dividend $(x^3-3x^2-7x+6)$ inside the division symbol and the divisor $(x-2)$ outside.
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Divide the leading terms. Divide the leading term of the dividend $(x^3)$ by the leading term of the divisor $(x)$. This gives us $x^2$. Write $x^2$ above the division symbol.
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Multiply the divisor by the quotient. Multiply $(x-2)$ by $x^2$ to get $x^3 - 2x^2$. Write this below the dividend.
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Subtract. Subtract the result from the dividend. This gives us $-x^2 - 7x$.
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Bring down the next term. Bring down the next term of the dividend (-7x).
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Repeat steps 2-5. Repeat the process, now dividing the new leading term $(-x^2)$ by the leading term of the divisor $(x)$. This gives us $-x$. Write $-x$ next to $x^2$ above the division symbol.
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Continue until the degree of the remainder is less than the degree of the divisor. Keep repeating steps 2-5 until the degree of the remaining polynomial is less than the degree of the divisor.
The Solution
Let's apply the steps to our problem:
x^2 - x - 9
x - 2 | x^3 - 3x^2 - 7x + 6
x^3 - 2x^2
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-x^2 - 7x
-x^2 + 2x
------------
-9x + 6
-9x + 18
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-12
Therefore, $(x^3-3x^2-7x+6) \div (x-2) = \boxed{x^2 - x - 9}$ with a remainder of -12.
This can be written as:
$x^3-3x^2-7x+6 = (x-2)(x^2-x-9) - 12$
Conclusion
Polynomial long division is a crucial technique for simplifying and manipulating polynomials. By applying the steps systematically, we can successfully divide any polynomial by another polynomial.