Find the product.
[tex](a^2)(2a^3)(a^2-8a+9)[/tex]
Answer (2)
Multiply a 2 and 2 a 3 to get 2 a 5 .
Distribute 2 a 5 over ( a 2 − 8 a + 9 ) .
Multiply each term: 2 a 5 ⋅ a 2 = 2 a 7 , 2 a 5 ⋅ ( − 8 a ) = − 16 a 6 , and 2 a 5 ⋅ 9 = 18 a 5 .
Combine the terms to get the final answer: 2 a 7 − 16 a 6 + 18 a 5 .
Explanation
Understanding the Problem Let's break down this problem. We are asked to find the product of three terms: a 2 , 2 a 3 , and ( a 2 − 8 a + 9 ) . This involves multiplying these terms together and simplifying the result.
Multiplying the First Two Terms First, let's multiply the first two terms: a 2 and 2 a 3 . When multiplying terms with the same base, we add their exponents. So, we have:
a 2 ⋅ 2 a 3 = 2 a 2 + 3 = 2 a 5
Distributing the Product Now, we need to multiply the result, 2 a 5 , by the third term, which is ( a 2 − 8 a + 9 ) . We'll use the distributive property, which means we multiply 2 a 5 by each term inside the parentheses:
2 a 5 ( a 2 − 8 a + 9 ) = 2 a 5 ⋅ a 2 − 2 a 5 ⋅ 8 a + 2 a 5 ⋅ 9
Simplifying the Expression Now, let's simplify each of these terms. Again, we add the exponents when multiplying terms with the same base:
2 a 5 ⋅ a 2 = 2 a 5 + 2 = 2 a 7
− 2 a 5 ⋅ 8 a = − 16 a 5 + 1 = − 16 a 6
2 a 5 ⋅ 9 = 18 a 5
So, combining these, we get:
2 a 7 − 16 a 6 + 18 a 5
Final Answer Therefore, the product of the given expression is 2 a 7 − 16 a 6 + 18 a 5 .
Examples
Understanding polynomial multiplication is crucial in various fields, such as physics and engineering, where complex systems are modeled using polynomial equations. For instance, when calculating the trajectory of a projectile, engineers use polynomial functions to represent the object's position over time. Multiplying polynomials helps in combining different factors affecting the trajectory, such as initial velocity, angle of launch, and gravitational forces, to predict the projectile's path accurately. This enables them to design systems and predict outcomes effectively.
To find the product ( a 2 ) ( 2 a 3 ) ( a 2 − 8 a + 9 ) , we first multiply the monomials to get 2 a 5 , then distribute it over the polynomial. The final combined expression is 2 a 7 − 16 a 6 + 18 a 5 .
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