# How do you divide #(4x^3 -12x^2+5x+3)/((x +3) )#?

polynomial long division - yay!

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To divide (4x^3 - 12x^2 + 5x + 3) by (x + 3), you can use long division or synthetic division. Here is the step-by-step process using long division:

- Divide the first term of the dividend (4x^3) by the first term of the divisor (x). This gives you 4x^2.
- Multiply the divisor (x + 3) by the quotient obtained in step 1 (4x^2). This gives you 4x^3 + 12x^2.
- Subtract the result obtained in step 2 from the dividend (4x^3 - 12x^2 + 5x + 3) to get a new polynomial: (-12x^2 + 5x + 3) - (4x^3 + 12x^2) = -4x^3 - 24x^2 + 5x + 3.
- Bring down the next term from the dividend, which is 5x.
- Divide the new polynomial (-4x^3 - 24x^2 + 5x + 3) by the first term of the divisor (x). This gives you -4x^2.
- Multiply the divisor (x + 3) by the quotient obtained in step 5 (-4x^2). This gives you -4x^3 - 12x^2.
- Subtract the result obtained in step 6 from the new polynomial (-4x^3 - 24x^2 + 5x + 3) to get a new polynomial: (-24x^2 + 5x + 3) - (-4x^3 - 12x^2) = 4x^3 - 12x^2 + 5x + 3.
- Bring down the next term from the dividend, which is 3.
- Divide the new polynomial (4x^3 - 12x^2 + 5x + 3) by the first term of the divisor (x). This gives you 4x.
- Multiply the divisor (x + 3) by the quotient obtained in step 9 (4x). This gives you 4x^2 + 12x.
- Subtract the result obtained in step 10 from the new polynomial (4x^3 - 12x^2 + 5x + 3) to get a new polynomial: (-12x^2 + 5x + 3) - (4x^2 + 12x) = -16x^2 - 7x + 3.

Therefore, the quotient is 4x^2 - 4x + 1, and the remainder is -16x^2 - 7x + 3.

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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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