Multiply the polynomials.
[tex](x-5)(x^2+4x-2)[/tex]
A. [tex]x^3-x^2-22x+10[/tex]
B. [tex]x^3+4x^2-20x+10[/tex]
C. [tex]x^3-x^2-20x+10[/tex]
D. [tex]x^3+4x^2-2x+10[/tex]
Answer (2)
Distribute ( x − 5 ) over ( x 2 + 4 x − 2 ) .
Multiply each term: x ( x 2 + 4 x − 2 ) − 5 ( x 2 + 4 x − 2 ) = x 3 + 4 x 2 − 2 x − 5 x 2 − 20 x + 10 .
Combine like terms: x 3 − x 2 − 22 x + 10 .
The result of the multiplication is x 3 − x 2 − 22 x + 10 .
Explanation
Understanding the Problem We are given two polynomials, ( x − 5 ) and ( x 2 + 4 x − 2 ) . Our goal is to multiply these polynomials and identify the correct result from the given options.
Applying the Distributive Property To multiply the polynomials, we'll use the distributive property. This means each term in the first polynomial is multiplied by each term in the second polynomial.
Expanding the Expression Here's how the multiplication looks:
( x − 5 ) ( x 2 + 4 x − 2 ) = x ( x 2 + 4 x − 2 ) − 5 ( x 2 + 4 x − 2 )
Distributing Now, let's distribute x and − 5 across the second polynomial:
x ( x 2 + 4 x − 2 ) = x 3 + 4 x 2 − 2 x
− 5 ( x 2 + 4 x − 2 ) = − 5 x 2 − 20 x + 10
Combining the Results Next, we combine these two results:
x 3 + 4 x 2 − 2 x − 5 x 2 − 20 x + 10
Combining Like Terms Now, we combine like terms:
x 3 + ( 4 x 2 − 5 x 2 ) + ( − 2 x − 20 x ) + 10
= x 3 − x 2 − 22 x + 10
Identifying the Correct Option Finally, we compare our result with the given options. The correct answer is:
A. x 3 − x 2 − 22 x + 10
Examples
Polynomial multiplication is used in various fields, such as engineering, physics, and computer graphics. For instance, 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 ensure the bridge's stability. Similarly, in computer graphics, polynomial multiplication is used to create smooth curves and surfaces, which are essential for realistic rendering.
To multiply the polynomials ( x − 5 ) ( x 2 + 4 x − 2 ) , we distribute each term and combine like terms to arrive at the expression x 3 − x 2 − 22 x + 10 . The correct answer is option A.
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