Multiply. $(3 x-5)(-4 x+7)$ Simplify your answer.
Answer (2)
To multiply the binomials ( 3 x − 5 ) ( − 4 x + 7 ) , we used FOIL, resulting in the simplified expression − 12 x 2 + 41 x − 35 .
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Multiply the first terms: ( 3 x ) ( − 4 x ) = − 12 x 2 .
Multiply the outer terms: ( 3 x ) ( 7 ) = 21 x .
Multiply the inner terms: ( − 5 ) ( − 4 x ) = 20 x .
Multiply the last terms: ( − 5 ) ( 7 ) = − 35 . Combine like terms: − 12 x 2 + 41 x − 35 .
Explanation
Understanding the Problem We are asked to multiply two binomials, ( 3 x − 5 ) and ( − 4 x + 7 ) , and simplify the result. This involves using the distributive property (often remembered by the acronym FOIL: First, Outer, Inner, Last) to expand the product and then combining like terms to simplify.
Multiplying the First Terms First, we multiply the first terms of each binomial: ( 3 x ) × ( − 4 x ) = − 12 x 2
Multiplying the Outer Terms Next, we multiply the outer terms: ( 3 x ) × ( 7 ) = 21 x
Multiplying the Inner Terms Then, we multiply the inner terms: ( − 5 ) × ( − 4 x ) = 20 x
Multiplying the Last Terms Finally, we multiply the last terms: ( − 5 ) × ( 7 ) = − 35
Combining the Terms Now, we combine all the terms we found: − 12 x 2 + 21 x + 20 x − 35
Simplifying the Expression We simplify by combining the like terms, which are 21 x and 20 x : 21 x + 20 x = 41 x So the simplified expression is: − 12 x 2 + 41 x − 35
Final Answer Therefore, the product of ( 3 x − 5 ) and ( − 4 x + 7 ) simplified is − 12 x 2 + 41 x − 35 .
Examples
Understanding how to multiply binomials is essential in various real-world applications, such as calculating areas, modeling growth, and solving optimization problems. For instance, if you're designing a rectangular garden where the length is ( 3 x − 5 ) feet and the width is ( − 4 x + 7 ) feet, multiplying these binomials gives you the area of the garden as a function of x . This allows you to determine the area for different values of x and optimize the garden's dimensions based on available space or resources. The ability to manipulate algebraic expressions like this is a fundamental skill in engineering, economics, and computer science.
