Use the table to find the products of the two polynomials. Write your answer in descending order.
A) $\left(4 x^2-4 x\right)\left(x^2-4\right)$
Answer (1)
Multiply x 2 by each term of ( 4 x 2 − 4 x ) : 4 x 4 − 4 x 3 .
Multiply − 4 by each term of ( 4 x 2 − 4 x ) : − 16 x 2 + 16 x .
Combine the terms: 4 x 4 − 4 x 3 − 16 x 2 + 16 x .
Write the final polynomial in descending order: 4 x 4 − 4 x 3 − 16 x 2 + 16 x .
Explanation
Understanding the Problem We are given the polynomials ( 4 x 2 − 4 x ) and ( x 2 − 4 ) , and we need to find their product and write the result in descending order. The table helps us organize the multiplication of each term.
Multiplying by x 2 First, we multiply x 2 by each term of the first polynomial: x 2 ∗ 4 x 2 = 4 x 4 x 2 ∗ ( − 4 x ) = − 4 x 3
Multiplying by -4 Next, we multiply -4 by each term of the first polynomial: − 4 ∗ 4 x 2 = − 16 x 2 − 4 ∗ ( − 4 x ) = 16 x
Combining the Terms Now, we add all the terms together: 4 x 4 − 4 x 3 − 16 x 2 + 16 x
Final Answer Finally, we write the polynomial in descending order, which it already is: 4 x 4 − 4 x 3 − 16 x 2 + 16 x
Examples
Polynomial multiplication is used in various fields such as engineering, physics, and computer science. For example, in control systems, the transfer function of a system can be represented as a rational function, which involves polynomial multiplication. In computer graphics, polynomial multiplication is used in curve and surface design. Understanding polynomial multiplication helps in modeling and analyzing these real-world systems.
