Questions: Rewrite without parentheses. -3 x^2(4 x^3-9 x-3) Simplify your answer as much as possible.

Transcript text: Rewrite without parentheses. \[ -3 x^{2}\left(4 x^{3}-9 x-3\right) \] Simplify your answer as much as possible.

Solution

Solution Steps

To solve the problem of multiplying a univariate polynomial by a monomial with a negative, we will distribute the monomial \(-3x^2\) across each term inside the parentheses. This involves multiplying \(-3x^2\) by each term in the polynomial \(4x^3 - 9x - 3\). After performing the multiplication, we will combine like terms if necessary to simplify the expression.

Step 1: Distributing the Monomial

We start with the expression \(-3x^2(4x^3 - 9x - 3)\). To simplify this, we distribute \(-3x^2\) across each term in the polynomial:

\[ -3x^2 \cdot 4x^3 = -12x^5 \] \[ -3x^2 \cdot (-9x) = 27x^3 \] \[ -3x^2 \cdot (-3) = 9x^2 \]

Step 2: Combining the Results

After distributing, we combine the results from the previous step:

\[ -12x^5 + 27x^3 + 9x^2 \]