Question

Which of the following expressions is equivalent to the expression above? A) \(4 x^6+3 x^4-3 x^3+2\) B) \(4 x^6+3 x^4+7 x^3+2\) C) \(4 x^6+3 x^4-5 x^3+2\) D) \(7 x^{10}-3 x^3+2\)

Short answer

Unfortunately, none of the given options match the simplified expression \(P(x) = 4x^6 + 2x^5 + 2x^4 - 5x^3 - 3x\).

Step-by-step solution

Identify the main operation in the given expression

The main operation in the given expression \(P(x)\) is multiplication. The expression is the product of two factors: \((2x^3 + x)\) and \((2x^3 + x^2 - 3)\).

Expand the expression

To expand the given expression, we will apply the distributive property to multiply each term in the first factor by each term in the second factor. It will look like this: \(P(x) = (2x^3 + x)(2x^3 + x^2 - 3)\\ = 2x^3(2x^3 + x^2 - 3) + x(2x^3 + x^2 - 3)\)

Distribute the terms and simplify

We'll perform the multiplication and addition operations to fully expand and simplify the expression. \(P(x) = 2x^3(2x^3 + x^2 - 3) + x(2x^3 + x^2 - 3) \\ = (4x^6 + 2x^5 - 6x^3) + (2x^4 + x^3 - 3x) \\ = 4x^6 + 2x^5 - 6x^3 + 2x^4 + x^3 - 3x\) Now, we will combine like terms: \(P(x) = 4x^6 + 2x^5 + 2x^4 - 5x^3 - 3x\)

Compare with the given options

Now let's compare our simplified expression with the listed options: A) \(4x^6 + 3x^4 - 3x^3 + 2 \\ B) 4x^6 + 3x^4 + 7x^3 + 2 \\ C) 4x^6 + 3x^4 - 5x^3 + 2 \\ D) 7x^{10} - 3x^3 + 2\) As we can observe, there are no matching options available. However, the correct method to approach this kind of problem has been demonstrated. If there was an option that matched the simplified expression after Step 3, that would be the correct answer.