Multiply and simplify: $-3 x^2(x^3+2 x-5)$
Answer (2)
Distribute − 3 x 2 to each term inside the parentheses: ( − 3 x 2 ) ( x 3 ) + ( − 3 x 2 ) ( 2 x ) + ( − 3 x 2 ) ( − 5 ) .
Simplify each term using the exponent rule x a ⋅ x b = x a + b .
Combine the simplified terms: − 3 x 5 − 6 x 3 + 15 x 2 .
The simplified expression is − 3 x 5 − 6 x 3 + 15 x 2 .
Explanation
Understanding the Problem We are given the expression − 3 x 2 ( x 3 + 2 x − 5 ) to multiply and simplify. This involves distributing the term − 3 x 2 to each term inside the parentheses.
Applying the Distributive Property We apply the distributive property to multiply − 3 x 2 by each term in the trinomial ( x 3 + 2 x − 5 ) . This gives us: ( − 3 x 2 ) ( x 3 ) + ( − 3 x 2 ) ( 2 x ) + ( − 3 x 2 ) ( − 5 )
Simplifying Each Term Now, we simplify each term. Recall the exponent rule x a ⋅ x b = x a + b .
For the first term: ( − 3 x 2 ) ( x 3 ) = − 3 x 2 + 3 = − 3 x 5 .
For the second term: ( − 3 x 2 ) ( 2 x ) = − 6 x 2 + 1 = − 6 x 3 .
For the third term: ( − 3 x 2 ) ( − 5 ) = 15 x 2 .
Combining the Terms Combining the simplified terms, we get: − 3 x 5 − 6 x 3 + 15 x 2
Final Answer Therefore, the simplified expression is − 3 x 5 − 6 x 3 + 15 x 2 .
Examples
Polynomial multiplication is used in various fields, such as physics and engineering, to model complex systems. For example, when calculating the trajectory of a projectile, you might use polynomial expressions to represent the initial velocity and angle, and multiplying these polynomials helps determine the projectile's path. Similarly, in economics, polynomial multiplication can be used to model revenue and cost functions to optimize profit.
To simplify − 3 x 2 ( x 3 + 2 x − 5 ) , we distribute − 3 x 2 across each term in the parentheses. The resulting expression simplifies to − 3 x 5 − 6 x 3 + 15 x 2 .
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