Multiply and simplify: $5 x^3\left(6 x^2+2 x-4\right)$

To simplify the expression 5 x 3 ( 6 x 2 + 2 x − 4 ) , distribute 5 x 3 to each term inside the parentheses. This results in the simplified expression 30 x 5 + 10 x 4 − 20 x 3 .
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Answered by Anonymous | 2025-07-04

Distribute 5 x 3 across the terms inside the parenthesis: 5 x 3 ( 6 x 2 + 2 x − 4 ) = 5 x 3 × 6 x 2 + 5 x 3 × 2 x − 5 x 3 × 4 .
Multiply the coefficients and add the exponents: 30 x 5 + 10 x 4 − 20 x 3 .
The simplified expression is: 30 x 5 + 10 x 4 − 20 x 3 ​ .

Explanation

Understanding the problem We are asked to multiply and simplify the expression 5 x 3 ( 6 x 2 + 2 x − 4 ) . This involves distributing the term 5 x 3 to each term inside the parentheses.

Distributing the term We distribute 5 x 3 to each term inside the parentheses:


5 x 3 ( 6 x 2 + 2 x − 4 ) = 5 x 3 ⋅ 6 x 2 + 5 x 3 ⋅ 2 x − 5 x 3 ⋅ 4

Multiplying the terms Now, we multiply the coefficients and add the exponents of x in each term:

5 x 3 ⋅ 6 x 2 = ( 5 ⋅ 6 ) x 3 + 2 = 30 x 5
5 x 3 ⋅ 2 x = ( 5 ⋅ 2 ) x 3 + 1 = 10 x 4
5 x 3 ⋅ 4 = ( 5 ⋅ 4 ) x 3 = 20 x 3

Combining the terms Combining these terms, we get the simplified expression:

30 x 5 + 10 x 4 − 20 x 3

Final Answer Therefore, the simplified expression is 30 x 5 + 10 x 4 − 20 x 3 .

Examples
Polynomial multiplication is used in various fields such as physics, engineering, and computer science. For example, when calculating the trajectory of a projectile, you might use polynomial equations to model the path. Multiplying polynomials helps in combining different factors affecting the trajectory, such as initial velocity and gravitational force, to predict where the projectile will land. Similarly, in computer graphics, polynomial multiplication is used to create smooth curves and surfaces by combining different polynomial functions.

Answered by GinnyAnswer | 2025-07-04