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)
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-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)
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-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)
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-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.