Simplify the following expression.
$(2 x-3)\left(5 x^4-7 x^3+6 x^2-9\right)$
Answer (1)
To simplify the expression ( 2 x − 3 ) ( 5 x 4 − 7 x 3 + 6 x 2 − 9 ) :
Multiply each term in the first factor by each term in the second factor.
Combine the like terms.
The simplified expression is 10 x 5 − 29 x 4 + 33 x 3 − 18 x 2 − 18 x + 27 .
The final answer is 10 x 5 − 29 x 4 + 33 x 3 − 18 x 2 − 18 x + 27 .
Explanation
Understanding the Problem We are given the expression ( 2 x − 3 ) ( 5 x 4 − 7 x 3 + 6 x 2 − 9 ) and four possible simplified expressions. Our goal is to expand the given expression and determine which of the four options matches our result.
Expanding the Expression To expand the expression, we'll multiply each term in the first factor ( 2 x − 3 ) by each term in the second factor ( 5 x 4 − 7 x 3 + 6 x 2 − 9 ) . This is an application of the distributive property.
Performing the Multiplication Let's perform the multiplication:
2 x ∗ ( 5 x 4 − 7 x 3 + 6 x 2 − 9 ) = 10 x 5 − 14 x 3 + 12 x 3 − 18 x − 3 ∗ ( 5 x 4 − 7 x 3 + 6 x 2 − 9 ) = − 15 x 4 + 21 x 3 − 18 x 2 + 27
Now, we add these two results together: ( 10 x 5 − 14 x 4 + 12 x 3 − 18 x ) + ( − 15 x 4 + 21 x 3 − 18 x 2 + 27 ) = 10 x 5 − 14 x 4 − 15 x 4 + 12 x 3 + 21 x 3 − 18 x 2 − 18 x + 27
Combining Like Terms Now, we combine like terms:
10 x 5 + ( − 14 x 4 − 15 x 4 ) + ( 12 x 3 + 21 x 3 ) − 18 x 2 − 18 x + 27 = 10 x 5 − 29 x 4 + 33 x 3 − 18 x 2 − 18 x + 27
Identifying the Correct Expression Comparing our expanded expression 10 x 5 − 29 x 4 + 33 x 3 − 18 x 2 − 18 x + 27 with the given options, we find that it matches the fourth option: 10 x 5 − 29 x 4 + 33 x 3 − 18 x 2 − 18 x + 27
Final Answer Therefore, the simplified expression is 10 x 5 − 29 x 4 + 33 x 3 − 18 x 2 − 18 x + 27 .
Examples
Polynomial multiplication is used in various fields such as engineering, physics, and computer graphics. For example, in computer graphics, polynomial multiplication can be used to combine different transformations (like scaling, rotation, and translation) represented as polynomials to create complex animations or visual effects. Understanding how to simplify and manipulate polynomial expressions is crucial for optimizing these processes and achieving desired outcomes.
