Which of the following options is an equivalent form of the rational expression 6x3+14x2+8x2x
?
Option #1: 3x2+7x+4
Option #2: 3x3+7x2+4x
Option #3: The rational expression has no equivalent forms.
The phrase "rational expression" is typically used in the context of ratios of polynomials. Here, the expression given (6x3+14x2+8x2x) is not a ratio, but simply a polynomial. Additionally, there seems to be a mistake in the expression -- 8x2x is not clear. If it's supposed to be 8x^3, then the polynomial simplifies to:
14x^3 + 14x^2 which is the same as 14x^2(x+1)
None of the options given actually match this. So perhaps the closest answer here would be:
Option #3: The rational expression has no equivalent forms.
To determine which option is an equivalent form of the given rational expression, you need to simplify the expression and compare it with the options provided.
The given rational expression is 6x^3 + 14x^2 + 8x^2x.
To simplify it, we can factor out the common term, which is x^2:
6x^3 + 14x^2 + 8x^2x
= x^2(6x + 14 + 8x)
= x^2(14x + 14)
Now, we can further simplify it by factoring out the common term, which is 14:
x^2(14x + 14)
= x^2 * 14(x + 1)
= 14x^2(x + 1)
Comparing the simplified expression, 14x^2(x + 1), with the options provided:
Option #1: 3x^2 + 7x + 4 (not equivalent)
Option #2: 3x^3 + 7x^2 + 4x (not equivalent)
Option #3: The rational expression has an equivalent form (not correct)
Therefore, the correct option is none of the provided options, as there is no exact equivalent form among them.
To determine which option is an equivalent form of the given rational expression, we need to simplify the expression.
However, there appears to be a mistake in the given expression. It should be the rational expression: 6x^3 + 14x^2 + 8x.
Now, let's factor out the common factor, if any, from all the terms:
6x^3 + 14x^2 + 8x = 2x(3x^2 + 7x + 4)
Now, let's factor the quadratic expression inside the brackets:
2x(3x^2 + 7x + 4) = 2x(3x + 4)(x + 1)
Therefore, the equivalent form of the given rational expression is:
Option #2: 3x^3 + 7x^2 + 4x
So, the correct answer is Option #2.