# How do you divide #(2x^3 - 7x^2 - 17x - 3) / (2x+3)#?

There are several ways to do this. I will give a brief demonstration of two of them:

synthetic division

#{: (,,2,-7,-17,-3), (+,,(0),-3,15,3), (,,"-----","-----","-----","-----"), (/(2),"|",2,-10,-2,color(blue)(0)), (color(white)("XX")xx(-3),"|",color(red)(1),color(red)(-5),color(red)(-1),) :}#

long polynomial division

#{: (,,color(red)(x^2),color(red)(-5x),color(red)(-1),), (,,"-----","-----","-----","-----"), (2x+3,")",2x^3,-7x^2,-17x,-3), (,,2x^3,+3x^2,,), (,,"-----","-----",,), (,,,-10x^2,-17x,), (,,,-10x^2,-15x,), (,,,"-----","-----",), (,,,,-2x,-3), (,,,,-2x,-3), (,,,,"-----","-----"), (,,,,,color(blue)(0)) :}#

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To divide (2x^3 - 7x^2 - 17x - 3) by (2x+3), you can use long division. Here are the steps:

- Divide the first term of the numerator (2x^3) by the first term of the denominator (2x). This gives x^2 as the first term of the quotient.
- Multiply the entire denominator (2x+3) by x^2, which gives 2x^3 + 3x^2.
- Subtract this result (2x^3 + 3x^2) from the numerator (2x^3 - 7x^2 - 17x - 3). This gives -10x^2 - 17x - 3.
- Bring down the next term from the numerator (-10x^2), and divide it by the first term of the denominator (2x). This gives -5x as the next term of the quotient.
- Multiply the entire denominator (2x+3) by -5x, which gives -10x^2 - 15x.
- Subtract this result (-10x^2 - 15x) from the previous result (-10x^2 - 17x - 3). This gives -2x - 3.
- Bring down the next term from the numerator (-2x), and divide it by the first term of the denominator (2x). This gives -1 as the next term of the quotient.
- Multiply the entire denominator (2x+3) by -1, which gives -2x - 3.
- Subtract this result (-2x - 3) from the previous result (-2x - 3). This gives 0.

The quotient is x^2 - 5x - 1, and since the remainder is 0, there are no more terms left to bring down.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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