To determine which polynomial divides evenly by \(x + 3\) using long division, we generally need to perform polynomial long division. However, you haven't provided any specific polynomials to test. Let's go through the process with a generic polynomial \(P(x)\) and a divisor \(D(x) = x + 3\).
Here is how the process of long division works:
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Setup the Division: Write \(P(x)\) (the polynomial you want to divide) under the long division symbol and \(x + 3\) (the divisor) outside.
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Divide the Leading Terms: Divide the leading term of \(P(x)\) by the leading term of \(D(x)\). This result will be the first term of the quotient.
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Multiply: Multiply \(D(x)\) by the term you just found in the quotient.
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Subtract: Subtract this product from \(P(x)\) to get a new polynomial.
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Repeat: Repeat the process with the new polynomial until the degree of the remaining polynomial is less than the degree of the divisor \(x + 3\).
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Determine Divisibility: If the remainder is zero, then \(P(x)\) divides evenly by \(x + 3\).
To demonstrate, let's say we want to check if a polynomial such as \(P(x) = x^2 + 3x + 4\) is divisible by \(x + 3\).
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Setup: \[ \text{Divide } x^2 + 3x + 4 \text{ by } x + 3 \]
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Divide the Leading Terms: \[ x^2 \div x = x \]
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Multiply: \[ x(x + 3) = x^2 + 3x \]
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Subtract: \[ (x^2 + 3x + 4) - (x^2 + 3x) = 4 \]
Now we have a remainder of \(4\). Since the remainder is not zero, \(x^2 + 3x + 4\) does not divide evenly by \(x + 3\).
If you provide specific polynomials, I can perform the long division for each and check for divisibility!
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