Expand and combine like terms.
$(2 x^4+3 x^3)(2 x^4-3 x^3)=$
$\square$
Answer (2)
Recognize the expression as a difference of squares: ( a + b ) ( a − b ) = a 2 − b 2 .
Apply the difference of squares formula: ( 2 x 4 + 3 x 3 ) ( 2 x 4 − 3 x 3 ) = ( 2 x 4 ) 2 − ( 3 x 3 ) 2 .
Expand each term: ( 2 x 4 ) 2 = 4 x 8 and ( 3 x 3 ) 2 = 9 x 6 .
Combine the terms to get the final expression: 4 x 8 − 9 x 6 .
Explanation
Understanding the Problem We are given the expression ( 2 x 4 + 3 x 3 ) ( 2 x 4 − 3 x 3 ) and asked to expand and combine like terms.
Recognizing the Difference of Squares Notice that the given expression is in the form of a difference of squares: ( a + b ) ( a − b ) , where a = 2 x 4 and b = 3 x 3 . Recall that ( a + b ) ( a − b ) = a 2 − b 2 .
Applying the Formula Applying the difference of squares formula, we have ( 2 x 4 + 3 x 3 ) ( 2 x 4 − 3 x 3 ) = ( 2 x 4 ) 2 − ( 3 x 3 ) 2 .
Expanding Each Term Now, we expand each term: ( 2 x 4 ) 2 = 2 2 ⋅ ( x 4 ) 2 = 4 x 4 ⋅ 2 = 4 x 8 and ( 3 x 3 ) 2 = 3 2 ⋅ ( x 3 ) 2 = 9 x 3 ⋅ 2 = 9 x 6 .
Combining Like Terms Combining these terms, we get the final expanded expression: 4 x 8 − 9 x 6 .
Final Answer Therefore, the expanded and combined expression is 4 x 8 − 9 x 6 .
Examples
Understanding how to expand and simplify expressions like this is fundamental in algebra and calculus. For example, if you are designing a bridge, you might use polynomial expressions to model the load distribution. Simplifying these expressions allows engineers to quickly assess the structural integrity and make necessary adjustments. This ensures the bridge can withstand various stresses, making it safe for public use. Expanding and combining like terms is a basic algebraic skill that is used in many real-world applications.
The expression ( 2 x 4 + 3 x 3 ) ( 2 x 4 − 3 x 3 ) can be expanded using the difference of squares formula. After applying the formula and calculating squared terms, the final expression becomes 4 x 8 − 9 x 6 .
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