Let's simplify the expression:

\[
-t^3(2 - 5t)(t^2 + t + 3)
\]

### Step 1: Distribute \(-t^3\) to each term inside \((2 - 5t)\)

\[
-t^3(2 - 5t) = -t^3 \cdot 2 + (-t^3 \cdot -5t)
\]
\[
= -2t^3 + 5t^4
\]

### Step 2: Multiply the result by \((t^2 + t + 3)\)
Now, distribute each term of \(-2t^3 + 5t^4\) to \((t^2 + t + 3)\):

#### Multiply \( -2t^3 \) by \( t^2 + t + 3 \):
\[
-2t^3 \cdot t^2 = -2t^5
\]
\[
-2t^3 \cdot t = -2t^4
\]
\[
-2t^3 \cdot 3 = -6t^3
\]

So, the result of multiplying \(-2t^3\) is:
\[
-2t^5 - 2t^4 - 6t^3
\]

#### Multiply \( 5t^4 \) by \( t^2 + t + 3 \):
\[
5t^4 \cdot t^2 = 5t^6
\]
\[
5t^4 \cdot t = 5t^5
\]
\[
5t^4 \cdot 3 = 15t^4
\]

So, the result of multiplying \(5t^4\) is:
\[
5t^6 + 5t^5 + 15t^4
\]

### Step 3: Combine like terms
Now, let's add up all the terms from both parts:

\[
5t^6 + 5t^5 + 15t^4 - 2t^5 - 2t^4 - 6t^3
\]

Simplify by combining like terms:
\[
5t^6 + (5t^5 - 2t^5) + (15t^4 - 2t^4) - 6t^3
\]
\[
= 5t^6 + 3t^5 + 13t^4 - 6t^3
\]

### Final Answer:
\[
5t^6 + 3t^5 + 13t^4 - 6t^3
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you handle polynomials with more variables?
2. What are the rules for distributing terms in polynomials?
3. How can you factor large polynomial expressions?
4. How do you find the degree of a polynomial?
5. What techniques are used to simplify expressions with negative exponents?

### Tip:
Always group similar terms carefully to avoid mistakes when combining terms in polynomial expressions.

Simplify the polynomial expression -t^3(2-5t)(t^2+t+3) using the distributive property step by step. Learn how to distribute terms and combine like terms in this algebraic process, ideal for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation