Multiply and add like terms: $(3 y-4)(2 y+5)$
Answer (1)
Use the distributive property (FOIL) to expand ( 3 y − 4 ) ( 2 y + 5 ) .
Multiply each term: ( 3 y ) ( 2 y ) + ( 3 y ) ( 5 ) + ( − 4 ) ( 2 y ) + ( − 4 ) ( 5 ) = 6 y 2 + 15 y − 8 y − 20 .
Combine like terms: 15 y − 8 y = 7 y .
The simplified expression is 6 y 2 + 7 y − 20 .
Explanation
Understanding the Problem We are given the expression ( 3 y − 4 ) ( 2 y + 5 ) and asked to multiply and simplify by combining like terms. This involves expanding the product of two binomials.
Expanding the Product We will use the distributive property (also known as the FOIL method) to expand the product of the two binomials:
( 3 y − 4 ) ( 2 y + 5 ) = ( 3 y ) ( 2 y ) + ( 3 y ) ( 5 ) + ( − 4 ) ( 2 y ) + ( − 4 ) ( 5 )
Simplifying Each Term Now, we simplify each term:
( 3 y ) ( 2 y ) = 6 y 2
( 3 y ) ( 5 ) = 15 y
( − 4 ) ( 2 y ) = − 8 y
( − 4 ) ( 5 ) = − 20
So, we have: 6 y 2 + 15 y − 8 y − 20
Combining Like Terms Next, we combine like terms (the y terms):
15 y − 8 y = 7 y
So, we have: 6 y 2 + 7 y − 20
Final Result The simplified expression is 6 y 2 + 7 y − 20 .
Therefore, ( 3 y − 4 ) ( 2 y + 5 ) = 6 y 2 + 7 y − 20 .
Examples
Understanding how to multiply binomials is essential in various fields, such as physics and engineering, where polynomial expressions are frequently used to model physical phenomena. For example, when calculating the area of a rectangular garden with sides ( 3 y − 4 ) and ( 2 y + 5 ) , the expanded form 6 y 2 + 7 y − 20 gives the area as a quadratic function of y . This allows for easy calculation of the garden's area for different values of y , which can be useful in optimizing garden design and resource allocation. Similarly, in projectile motion, the height and range of a projectile can be expressed as polynomial functions, and multiplying these polynomials helps in analyzing the trajectory and impact point.
