Multiply the two polynomials using the distributive property.
A) $\left(4 x^2-4 x\right)\left(x^2-4\right)$
Answer (2)
Distribute 4 x 2 over ( x 2 − 4 ) to get 4 x 4 − 16 x 2 .
Distribute − 4 x over ( x 2 − 4 ) to get − 4 x 3 + 16 x .
Combine the results: ( 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 given two polynomials, ( 4 x 2 − 4 x ) and ( x 2 − 4 ) , and we need to multiply them using the distributive property.
Applying the Distributive Property To multiply these polynomials, we'll distribute each term of the first polynomial over the terms of the second polynomial. This means we'll multiply 4 x 2 and − 4 x by both x 2 and − 4 .
Distributing the First Term First, let's distribute 4 x 2 over ( x 2 − 4 ) : 4 x 2 ( x 2 − 4 ) = 4 x 2 ⋅ x 2 − 4 x 2 ⋅ 4 = 4 x 4 − 16 x 2
Distributing the Second Term Next, let's distribute − 4 x over ( x 2 − 4 ) : − 4 x ( x 2 − 4 ) = − 4 x ⋅ x 2 − 4 x ⋅ ( − 4 ) = − 4 x 3 + 16 x
Combining the Results Now, we combine the results from the two distributions: ( 4 x 4 − 16 x 2 ) + ( − 4 x 3 + 16 x ) = 4 x 4 − 4 x 3 − 16 x 2 + 16 x
Final Answer So, the final result of multiplying the two polynomials is: 4 x 4 − 4 x 3 − 16 x 2 + 16 x
Examples
Polynomial multiplication, like the one we just did, is used in various fields such as engineering, physics, and computer graphics. For example, when designing a bridge, engineers use polynomials to model the load and stress distribution. Multiplying these polynomials helps them understand how different factors interact and ensures the bridge's stability. Similarly, in computer graphics, polynomial multiplication is used to create smooth curves and surfaces.
To multiply the polynomials ( 4 x 2 − 4 x ) ( x 2 − 4 ) , we distribute each term in the first polynomial across both terms in the second polynomial. The final result is 4 x 4 − 4 x 3 − 16 x 2 + 16 x . This demonstrates the distributive property of multiplication with polynomials.
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