Multiply: [tex](3 x)(x^4+5)[/tex]
Answer (2)
Distribute the term ( 3 x ) to both terms inside the parenthesis ( x 4 + 5 ) .
Multiply ( 3 x ) by x 4 : 3 x ⋅ x 4 = 3 x 5 .
Multiply ( 3 x ) by 5 : 3 x ⋅ 5 = 15 x .
Add the two terms obtained in the previous steps: 3 x 5 + 15 x . The final answer is 3 x 5 + 15 x .
Explanation
Understanding the Problem We are asked to multiply the polynomial ( 3 x ) by the polynomial ( x 4 + 5 ) . This involves distributing the term 3 x to each term inside the parentheses.
Solution Plan To multiply ( 3 x ) by ( x 4 + 5 ) , we distribute 3 x to both x 4 and 5 . This means we perform the following operations:
Multiply ( 3 x ) by x 4 .
Multiply ( 3 x ) by 5 .
Add the results together.
Multiplying the First Term First, let's multiply ( 3 x ) by x 4 . When multiplying terms with exponents, we add the exponents. So, 3 x ⋅ x 4 = 3 x 1 + 4 = 3 x 5 .
Multiplying the Second Term Next, let's multiply ( 3 x ) by 5 . This is a straightforward multiplication: 3 x ⋅ 5 = 15 x .
Combining the Terms Now, we add the results from the previous two steps: 3 x 5 + 15 x . This is the final result of the multiplication.
Final Answer Therefore, the product of ( 3 x ) and ( x 4 + 5 ) is 3 x 5 + 15 x .
Examples
Polynomial multiplication is a fundamental concept in algebra and is used in various real-world applications. For example, engineers use polynomial multiplication to model the behavior of systems, such as the trajectory of a projectile or the stress on a bridge. In computer graphics, polynomial multiplication is used to perform transformations on objects, such as scaling, rotation, and translation. Understanding polynomial multiplication is essential for solving many problems in science, engineering, and computer science.
To multiply ( 3 x ) ( x 4 + 5 ) , distribute 3 x to both terms inside the parentheses, resulting in 3 x 5 + 15 x after performing the multiplications. This process is called the distributive property and is fundamental in algebra. Therefore, the final answer is 3 x 5 + 15 x .
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