$\begin{array}{r}4 x^2-4 x \\ \times \quad x^2-4 \\ \hline\end{array}$
Answer (1)
Multiply 4 x 2 by ( x 2 − 4 ) to get 4 x 4 − 16 x 2 .
Multiply − 4 x by ( x 2 − 4 ) to get − 4 x 3 + 16 x .
Add the two resulting expressions: ( 4 x 4 − 16 x 2 ) + ( − 4 x 3 + 16 x ) .
The final result is 4 x 4 − 4 x 3 − 16 x 2 + 16 x .
Explanation
Understanding the Problem We are asked to multiply two polynomials: ( 4 x 2 − 4 x ) and ( x 2 − 4 ) . This is a straightforward application of the distributive property.
Setting up the Multiplication We will multiply each term of the first polynomial by each term of the second polynomial. This is often visualized using the distributive property (also known as the FOIL method when multiplying two binomials, though here one of our expressions is a trinomial).
Multiplying the First Term First, multiply 4 x 2 by ( x 2 − 4 ) : 4 x 2 ( x 2 − 4 ) = 4 x 2 ⋅ x 2 − 4 x 2 ⋅ 4 = 4 x 4 − 16 x 2
Multiplying the Second Term Next, multiply − 4 x by ( x 2 − 4 ) : − 4 x ( x 2 − 4 ) = − 4 x ⋅ x 2 − 4 x ⋅ ( − 4 ) = − 4 x 3 + 16 x
Combining Like Terms Now, add the two resulting expressions: ( 4 x 4 − 16 x 2 ) + ( − 4 x 3 + 16 x ) = 4 x 4 − 4 x 3 − 16 x 2 + 16 x
Final Result Therefore, the final result of the multiplication 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, when designing a bridge, engineers use polynomial functions to model the load and stress distribution. Multiplying these polynomials helps them understand the combined effect of different factors on the bridge's structural integrity. Similarly, in computer graphics, polynomial multiplication is used to perform transformations and rendering of 3D objects.
