Multiply.
$\left(a^2+6\right)(5 a+1)=$
$\square$
Answer (2)
Expand the expression using the distributive property: ( a 2 + 6 ) ( 5 a + 1 ) = a 2 ( 5 a + 1 ) + 6 ( 5 a + 1 ) .
Distribute a 2 and 6 : a 2 ( 5 a + 1 ) = 5 a 3 + a 2 and 6 ( 5 a + 1 ) = 30 a + 6 .
Combine the expanded terms: 5 a 3 + a 2 + 30 a + 6 .
The final expanded form is: 5 a 3 + a 2 + 30 a + 6 .
Explanation
Understanding the Problem We are asked to multiply the expression ( a 2 + 6 ) ( 5 a + 1 ) . This involves using the distributive property to expand the product of the two terms.
Applying the Distributive Property We will multiply each term in the first parenthesis by each term in the second parenthesis: ( a 2 + 6 ) ( 5 a + 1 ) = a 2 ( 5 a + 1 ) + 6 ( 5 a + 1 ) Now, we distribute a 2 and 6 :
a 2 ( 5 a + 1 ) = 5 a 3 + a 2 6 ( 5 a + 1 ) = 30 a + 6
Combining Like Terms Now, we combine the expanded terms: 5 a 3 + a 2 + 30 a + 6
Final Answer The expression is now fully expanded and simplified. The final expression is: 5 a 3 + a 2 + 30 a + 6 So, the answer is 5 a 3 + a 2 + 30 a + 6 .
Examples
Understanding polynomial multiplication is essential in various fields, such as physics and engineering, where complex systems are modeled using polynomial equations. For instance, when analyzing the trajectory of a projectile, engineers use polynomial functions to describe its path. Multiplying polynomials helps in combining different factors affecting the trajectory, such as initial velocity and gravitational forces, to predict the projectile's landing point accurately. This ensures that designs are precise and safe, demonstrating the practical importance of polynomial multiplication in real-world applications.
To multiply the expression ( a 2 + 6 ) ( 5 a + 1 ) , we use the distributive property to expand it into 5 a 3 + a 2 + 30 a + 6 . The final result is 5 a 3 + a 2 + 30 a + 6 .
;
