\begin{array}{r}x^2+x-2 \\ \times \quad 4 x^2-8 x \\ \hline\end{array}
Answer (1)
Multiply 4 x 2 by ( x 2 + x − 2 ) to get 4 x 4 + 4 x 3 − 8 x 2 .
Multiply − 8 x by ( x 2 + x − 2 ) to get − 8 x 3 − 8 x 2 + 16 x .
Add the resulting polynomials: ( 4 x 4 + 4 x 3 − 8 x 2 ) + ( − 8 x 3 − 8 x 2 + 16 x ) .
Combine like terms to obtain the final result: 4 x 4 − 4 x 3 − 16 x 2 + 16 x .
Explanation
Understanding the Problem We are asked to multiply two polynomials: x 2 + x − 2 and 4 x 2 − 8 x . This problem involves polynomial multiplication, which means we need to distribute each term of one polynomial across the terms of the other polynomial and then simplify by combining like terms.
Multiplying by 4 x 2 First, we multiply 4 x 2 by each term in ( x 2 + x − 2 ) : 4 x 2 × ( x 2 + x − 2 ) = 4 x 4 + 4 x 3 − 8 x 2
Multiplying by − 8 x Next, we multiply − 8 x by each term in ( x 2 + x − 2 ) : − 8 x × ( x 2 + x − 2 ) = − 8 x 3 − 8 x 2 + 16 x
Adding the Polynomials Now, we add the two resulting polynomials: ( 4 x 4 + 4 x 3 − 8 x 2 ) + ( − 8 x 3 − 8 x 2 + 16 x ) = 4 x 4 + ( 4 x 3 − 8 x 3 ) + ( − 8 x 2 − 8 x 2 ) + 16 x
Combining Like Terms Finally, we combine like terms to simplify the expression: 4 x 4 − 4 x 3 − 16 x 2 + 16 x
Final Result The result of multiplying ( x 2 + x − 2 ) by ( 4 x 2 − 8 x ) 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 polynomial functions helps them understand the combined effect of different factors and ensure the bridge's structural integrity. Similarly, in computer graphics, polynomial multiplication is used to perform transformations and rendering of 3D objects.
