b. [tex]c=-\frac{1}{3}\left[b+\frac{1}{a}(v-1)\right][/tex]
25. Find the product of
([tex](\operatorname{ard} \sqrt{y}-3 y) (3 y+2 \sqrt{y})[/tex]
Answer (2)
Expand the product: ( y − 3 y ) ( 3 y + 2 y ) = ( y ) ( 3 y ) + ( y ) ( 2 y ) + ( − 3 y ) ( 3 y ) + ( − 3 y ) ( 2 y ) .
Simplify each term: 3 y 3/2 + 2 y − 9 y 2 − 6 y 3/2 .
Combine like terms: − 3 y 3/2 + 2 y − 9 y 2 .
Rearrange the terms: − 9 y 2 − 3 y 3/2 + 2 y .
Explanation
Understanding the Problem We are asked to find the product of ( y − 3 y ) ( 3 y + 2 y ) . This involves expanding the expression and simplifying the terms.
Expanding the Product Let's expand the product using the distributive property (also known as the FOIL method): ( y − 3 y ) ( 3 y + 2 y ) = ( y ) ( 3 y ) + ( y ) ( 2 y ) + ( − 3 y ) ( 3 y ) + ( − 3 y ) ( 2 y )
Simplifying the Terms Now, let's simplify each term:
( y ) ( 3 y ) = 3 y y = 3 y 3/2
( y ) ( 2 y ) = 2 y
( − 3 y ) ( 3 y ) = − 9 y 2
( − 3 y ) ( 2 y ) = − 6 y y = − 6 y 3/2
Combining Like Terms Combine the simplified terms: 3 y 3/2 + 2 y − 9 y 2 − 6 y 3/2 Now, combine like terms (the terms with y 3/2 ): ( 3 y 3/2 − 6 y 3/2 ) + 2 y − 9 y 2 = − 3 y 3/2 + 2 y − 9 y 2
Final Result Rearrange the terms to present the polynomial in descending order of exponents: − 9 y 2 − 3 y 3/2 + 2 y
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 analyzing the trajectory of a projectile, engineers use polynomial functions to describe its path. Multiplying such functions helps predict the projectile's behavior under different conditions, ensuring accurate designs and safety measures. This algebraic manipulation allows for precise calculations and informed decision-making in real-world applications.
To find the product of ( y − 3 y ) ( 3 y + 2 y ) , we expanded the expression using the distributive property and simplified the terms. The resulting expression is − 9 y 2 − 3 y 3/2 + 2 y . This demonstrates the process of polynomial multiplication and simplification.
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