Would x + 3 be a factor of 3x^3+ 5x^2 2x - 12?
Ans: No, because there is a remainder of -54.
correct
To determine if a polynomial is a factor of another polynomial, we can use the Remainder Theorem. According to the Remainder Theorem, if we divide the polynomial 3x^3 + 5x^2 + 2x - 12 by the polynomial x + 3, and the remainder is zero, then x + 3 is a factor of 3x^3 + 5x^2 + 2x - 12.
Let's perform the division to see if there is a remainder.
Dividing 3x^3 + 5x^2 + 2x - 12 by x + 3, we get:
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x + 3 | 3x^3 + 5x^2 + 2x - 12
- (3x^3 + 9x^2)
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-4x^2 + 2x - 12
- (-4x^2 - 12x)
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14x - 12
- (14x + 42)
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-54
As we can see, there is a remainder of -54. Therefore, x + 3 is not a factor of 3x^3 + 5x^2 + 2x - 12.
To determine if x + 3 is a factor of 3x^3 + 5x^2 + 2x - 12, we can use the Remainder Theorem. The Remainder Theorem states that if a polynomial f(x) is divided by x - c, the remainder is equal to f(c).
In this case, we want to divide 3x^3 + 5x^2 + 2x - 12 by x + 3. So, we substitute c = -3 into the polynomial:
f(-3) = 3(-3)^3 + 5(-3)^2 + 2(-3) - 12
= 3(-27) + 5(9) - 6 - 12
= -81 + 45 - 6 - 12
= -54
The result, -54, is the remainder. Since the remainder is not equal to zero, we can conclude that x + 3 is not a factor of 3x^3 + 5x^2 + 2x - 12.