Multiply the polynomials $(2 x+3)(4 x-5)$
Answer (2)
Multiply the first terms: 2 x × 4 x = 8 x 2 .
Multiply the outer terms: 2 x × − 5 = − 10 x .
Multiply the inner terms: 3 × 4 x = 12 x .
Multiply the last terms: 3 × − 5 = − 15 . Combine like terms: 8 x 2 − 10 x + 12 x − 15 = 8 x 2 + 2 x − 15 . The final answer is 8 x 2 + 2 x − 15 .
Explanation
Understanding the Problem We are given two polynomials ( 2 x + 3 ) and ( 4 x − 5 ) . Our goal is to multiply these polynomials and simplify the result.
Applying the Distributive Property To multiply the polynomials, we use the distributive property (also known as the FOIL method). This means each term in the first polynomial is multiplied by each term in the second polynomial.
Performing the Multiplication ( 2 x + 3 ) ( 4 x − 5 ) = 2 x ( 4 x ) + 2 x ( − 5 ) + 3 ( 4 x ) + 3 ( − 5 ) Now, we perform each multiplication: 2 x ( 4 x ) = 8 x 2 2 x ( − 5 ) = − 10 x 3 ( 4 x ) = 12 x 3 ( − 5 ) = − 15
Combining Like Terms Substitute these results back into the expression: 8 x 2 − 10 x + 12 x − 15 Now, combine the like terms (the terms with the same power of x ): − 10 x + 12 x = 2 x So the expression becomes: 8 x 2 + 2 x − 15
Final Result The final result of multiplying the polynomials ( 2 x + 3 ) and ( 4 x − 5 ) is 8 x 2 + 2 x − 15 .
Examples
Polynomial multiplication is a fundamental skill in algebra and is used in various real-world applications. For example, if you are designing a rectangular garden where the length is ( 4 x − 5 ) meters and the width is ( 2 x + 3 ) meters, multiplying these two polynomials will give you the area of the garden in terms of x . This allows you to calculate the area for different values of x , helping you plan the layout and resource allocation for your garden.
To multiply the polynomials ( 2 x + 3 ) ( 4 x − 5 ) , we apply the distributive property (FOIL method) and find that the result is 8 x 2 + 2 x − 15 . This involves multiplying the first, outer, inner, and last terms and then combining the like terms. Ultimately, the final answer is 8 x 2 + 2 x − 15 .
;
