Using long division, determine which of the following polynomials divides evenly for a divisor of x+4
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To determine which of the polynomials divides evenly for a divisor of x + 4, we can use long division.
Let's start with the first polynomial:
x^3 + 3x^2 - 4x - 12
To divide this polynomial by x + 4, we first divide the highest degree term:
x^3 ÷ (x + 4) = x^2
Now, we multiply x + 4 by x^2:
x^2 * (x + 4) = x^3 + 4x^2
Next, we subtract this from the original polynomial:
(x^3 + 3x^2 - 4x - 12) - (x^3 + 4x^2) = -x^2 - 4x - 12
Now, we bring down the next term, which is -x^2:
- x^2 - 4x - 12
To continue the division, we divide -x^2 by x + 4:
- x^2 ÷ (x + 4) = -1x
Now, we multiply x + 4 by -1x:
-1x * (x + 4) = -x^2 - 4x
Next, we subtract this from the previous result:
(-x^2 - 4x - 12) - (-x^2 - 4x) = -12
As there are no more terms to bring down, the division is finished. The quotient is x^2 - 1x and the remainder is -12.
Therefore, the polynomial x^3 + 3x^2 - 4x - 12 does not divide evenly for a divisor of x + 4.
To determine if a polynomial divides evenly for a divisor of x+4, we need to perform long division. Let's say the polynomial we want to divide is called P(x), and the divisor is x+4.
1. Arrange P(x) in descending order of powers of x.
2. Start by dividing the highest power term of P(x) by the highest power term of the divisor (x+4).
- Write the result as the first term of the quotient. Let's call it Q(x).
3. Multiply the divisor (x+4) by the first term of the quotient (Q(x)).
- Subtract the result from P(x) and write the remainder.
4. Bring down the next term of P(x) to form a new dividend.
- This will include the remainder obtained in the previous step.
5. Divide the highest power term of the new dividend by the highest power term of the divisor.
- Write the result as the next term of the quotient.
6. Repeat steps 3-5 until we reach a remainder of zero or the degree of the remainder is less than the divisor's degree.
If we reach a remainder of zero, it means that the polynomial divides evenly, and the quotient obtained is the result.
If the remainder is non-zero, it means that the polynomial does not divide evenly.
Without knowing the specific polynomial, we cannot provide a detailed step-by-step solution. However, if you provide the polynomial, I can help you with the long division process.
To determine if a polynomial divides evenly by a given divisor using long division, you need to perform the following steps:
1. Write the dividend (the polynomial to be divided) in descending order of powers of x. For example, if the dividend is 3x^3 + 2x^2 - 5x + 7, write it as 3x^3 + 2x^2 - 5x + 7.
2. Write the divisor (x + 4) on the left side of the division symbol.
3. Divide the highest power term in the dividend by the highest power term in the divisor. In this case, divide 3x^3 by x. The result is 3x^2.
4. Multiply the entire divisor (x + 4) by the result obtained in step 3 (3x^2). Place the product (3x^3 + 12x^2) beneath the dividend, aligning like terms.
3x^2
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x + 4 | 3x^3 + 2x^2 - 5x + 7
- (3x^3 + 12x^2)
__________________
-10x^2 - 5x
5. Subtract the product (3x^3 + 12x^2) from the corresponding terms in the dividend. In this case, subtract (3x^2 + 12x^2) from (2x^2 - 5x). The result is -10x^2 - 5x.
6. Repeat steps 3 to 5 with the next term in the dividend, which is -10x^2. Divide -10x^2 by x, which gives -10x. Multiply the entire divisor (x + 4) by -10x and place the product (-10x^2 - 40x) beneath the previous result.
3x^2 - 10x
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x + 4 | 3x^3 + 2x^2 - 5x + 7
- (3x^3 + 12x^2)
__________________
-10x^2 - 5x
+ ( -10x^2 - 40x)
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-45x + 7
7. Subtract the product (-10x^2 - 40x) from the corresponding terms in the dividend. In this case, subtract (-10x - 40x) from (-5x + 7). The result is -45x + 7.
8. Repeat steps 3 to 5 with the next term in the dividend, which is -45x. Divide -45x by x, which gives -45. Multiply the entire divisor (x + 4) by -45 and place the product (-45x - 180) beneath the previous result.
3x^2 - 10x - 45
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x + 4 | 3x^3 + 2x^2 - 5x + 7
- (3x^3 + 12x^2)
__________________
-10x^2 - 5x
+ ( -10x^2 - 40x)
__________________
-45x + 7
+ ( -45x - 180)
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187
9. Subtract the product (-45x - 180) from the corresponding terms in the dividend. In this case, subtract (-45 - 180) from (7). The result is 187.
10. The remainder at this point is 187. If the remainder is zero, the polynomial divides evenly by the divisor. If the remainder is nonzero, the polynomial does not divide evenly.
In this case, the remainder is 187, which is nonzero. Therefore, the polynomial 3x^3 + 2x^2 - 5x + 7 does not divide evenly by x + 4.