Using long division determine which of the fallowing polynomials divides evenly for a divisor of x + 3
To determine if a polynomial divides evenly by x + 3 using long division, we divide the polynomial by x + 3 and check if the remainder is zero.
Let's take an example.
Suppose we have the polynomial 2x^3 + 5x^2 - 4x - 8 and we want to check if it divides evenly by x + 3.
-2x^2 + 11x + 29
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x + 3 | 2x^3 + 5x^2 - 4x - 8
1. We start by dividing the leading term of the dividend (2x^3) by the leading term of the divisor (x). This gives us 2x^2.
2. We multiply the entire divisor (x + 3) by the result from step 1 (2x^2) and write it under the dividend.
This gives us 2x^2(x + 3) = 2x^3 + 6x^2.
3. We subtract this product from the dividend: (2x^3 + 5x^2 - 4x - 8) - (2x^3 + 6x^2). Simplifying, we get -x^2 - 4x - 8.
4. We bring down the next term, which is -4x.
5. We repeat steps 1-4 with the new dividend (-x^2 - 4x - 8).
- Divide (-x^2) by x to get -x.
- Multiply (x + 3) by -x to get -x(x + 3) = -x^2 - 3x.
- Subtract this product from the new dividend: (-x^2 - 4x - 8) - (-x^2 - 3x). Simplifying, we get -x - 8.
6. Bring down the next term, which is -8.
7. Repeat steps 1-4 with the new dividend (-x - 8).
- Divide (-x) by x to get -1.
- Multiply (x + 3) by -1 to get -(x + 3) = -x - 3.
- Subtract this product from the new dividend: (-x - 8) - (-(x + 3)). Simplifying, we get -5.
8. We have reached the end of the polynomial, so we can say that 2x^3 + 5x^2 - 4x - 8 divided by x + 3 leaves a remainder of -5.
Since the remainder is not zero, the polynomial 2x^3 + 5x^2 - 4x - 8 does not divide evenly by x + 3.
To determine if a polynomial divides evenly by another polynomial, we can use long division. Let's use long division to check which of the following polynomials divides evenly by the divisor x + 3.
Let's start with the first polynomial:
1. x^3 + 2x^2 + x - 6 divided by x + 3.
We compare the highest power term of the dividend, which is x^3, with the highest power term of the divisor, which is x. To make them match, we divide x^3 by x, which gives us x^2. We write x^2 above the line.
x^2
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x + 3 | x^3 + 2x^2 + x - 6
Next, we multiply the divisor (x + 3) by the x^2 on top: x^2 * (x + 3) = x^3 + 3x^2. We subtract this result from the dividend.
x^2
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x + 3 | x^3 + 2x^2 + x - 6
- (x^3 + 3x^2)
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- x^2 + x
Now, we bring down the next term in the dividend, which is -6x. We compare it with the divisor (x + 3). To make them match, we divide -x^2 by x, which gives us -x. We write -x above the line.
x^2 - x
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x + 3 | x^3 + 2x^2 + x - 6
- (x^3 + 3x^2)
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- x^2 + x
-(- x^2 - 3x)
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4x - 6
We now bring down the next term in the dividend, which is 4x. We compare it with the divisor (x + 3). To make them match, we divide 4x by x, which gives us 4. We write 4 above the line.
x^2 - x + 4
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x + 3 | x^3 + 2x^2 + x - 6
- (x^3 + 3x^2)
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- x^2 + x
-(- x^2 - 3x)
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4x - 6
- (4x + 12)
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-18
Since we have no more terms in the dividend, we look at the remainder. In this case, the remainder is -18.
Therefore, the first polynomial x^3 + 2x^2 + x - 6 does not divide evenly by the divisor x + 3.
Please let me know if I can help you with anything else.
To determine if a polynomial divides evenly by another polynomial using long division, follow these steps:
Step 1: Write the dividend polynomial and the divisor polynomial in descending order, with all the terms in their appropriate places. Make sure to include any missing terms with a coefficient of zero.
Step 2: Divide the first term of the dividend polynomial by the first term of the divisor polynomial. Write the result above the long division symbol.
Step 3: Multiply the divisor polynomial by the quotient obtained in step 2. Write the result below the dividend polynomial, aligning like terms.
Step 4: Subtract the product obtained in step 3 from the dividend polynomial.
Step 5: Bring down the next term from the dividend polynomial.
Step 6: Repeat steps 2-5 until all terms from the dividend polynomial have been used.
Step 7: If there is no remainder and all terms have been used, then the divisor divides evenly into the dividend. If there is a remainder or any terms are unused, then the divisor does not divide evenly.
Now let's apply these steps to determine which polynomial divides evenly for a divisor of x + 3.
Let's say we have a dividend polynomial D(x) and a divisor polynomial d(x) = x + 3.
Perform long division as follows:
Step 1: Write the dividend polynomial and the divisor polynomial in descending order:
D(x) = ...
Step 2: Divide the first term of the dividend polynomial by the first term of the divisor polynomial. Write the result above the long division symbol.
Step 3: Multiply the divisor polynomial by the quotient obtained in step 2. Write the result below the dividend polynomial, aligning like terms.
Step 4: Subtract the product obtained in step 3 from the dividend polynomial.
Step 5: Bring down the next term from the dividend polynomial.
Step 6: Repeat steps 2-5 until all terms from the dividend polynomial have been used.
Step 7: Analyze the result. If there is no remainder and all terms have been used, then the divisor divides evenly into the dividend.
Following these steps will help you determine if a given polynomial divides evenly by another polynomial using long division.