What is the product?
$(4 s+2)\left(5 s^2+10 s+3\right)$
A. $20 s^2+20 s+6$
B. $20 s^3+40 s^2+12 s$
C. $20 s^3+10 s^2+32 s+6$
D. $20 s^3+50 s^2+32 s+6$
Answer (2)
The product of the expression ( 4 s + 2 ) ( 5 s 2 + 10 s + 3 ) is 20 s 3 + 50 s 2 + 32 s + 6 . Therefore, the correct answer is option D. This was obtained by distributing each term in the first polynomial across the second polynomial and combining like terms.
;
Distribute 4 s over ( 5 s 2 + 10 s + 3 ) to get 20 s 3 + 40 s 2 + 12 s .
Distribute 2 over ( 5 s 2 + 10 s + 3 ) to get 10 s 2 + 20 s + 6 .
Add the two resulting expressions: ( 20 s 3 + 40 s 2 + 12 s ) + ( 10 s 2 + 20 s + 6 ) .
Combine like terms to obtain the final product: 20 s 3 + 50 s 2 + 32 s + 6 .
Explanation
Understanding the Problem We are given the expression ( 4 s + 2 ) ( 5 s 2 + 10 s + 3 ) and asked to find the product.
Plan of Action To find the product, we will use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.
Distributing the First Term First, distribute 4 s over the second polynomial: 4 s ( 5 s 2 + 10 s + 3 ) = 4 s ∗ 5 s 2 + 4 s ∗ 10 s + 4 s ∗ 3 = 20 s 3 + 40 s 2 + 12 s
Distributing the Second Term Next, distribute 2 over the second polynomial: 2 ( 5 s 2 + 10 s + 3 ) = 2 ∗ 5 s 2 + 2 ∗ 10 s + 2 ∗ 3 = 10 s 2 + 20 s + 6
Combining the Results Now, add the two resulting expressions together: ( 20 s 3 + 40 s 2 + 12 s ) + ( 10 s 2 + 20 s + 6 ) = 20 s 3 + ( 40 s 2 + 10 s 2 ) + ( 12 s + 20 s ) + 6
Simplifying the Expression Combine like terms to simplify the expression: 20 s 3 + 50 s 2 + 32 s + 6
Final Answer Therefore, the product of the given polynomials is 20 s 3 + 50 s 2 + 32 s + 6 .
Examples
Polynomial multiplication is a fundamental concept in algebra and has many real-world applications. For example, engineers use polynomial multiplication to model the behavior of systems, such as the trajectory of a projectile or the growth of a population. In economics, polynomial multiplication can be used to model cost and revenue functions to optimize business decisions. Understanding polynomial multiplication is essential for solving complex problems in various fields.
