Multiply $(x-4)(x^2-3x+5)$.
A. $x^3+5x^2+5x-20$
B. $x^3+2x^2+8x-20$
C. $x^3-x^2+7x-20$
D. $x^3-7x^2+17x-20$

The product of ( x − 4 ) ( x 2 − 3 x + 5 ) is x 3 − 7 x 2 + 17 x − 20 , which is option D. The solution involved distributing each term in the first polynomial by each term in the second and combining like terms. The final result is obtained from careful calculation and simplification of polynomial expressions.
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Answered by Anonymous | 2025-07-04

Multiply x by ( x 2 − 3 x + 5 ) to get x 3 − 3 x 2 + 5 x .
Multiply − 4 by ( x 2 − 3 x + 5 ) to get − 4 x 2 + 12 x − 20 .
Add the two resulting polynomials: ( x 3 − 3 x 2 + 5 x ) + ( − 4 x 2 + 12 x − 20 ) .
Combine like terms to get the final answer: x 3 − 7 x 2 + 17 x − 20 ​ .

Explanation

Understanding the Problem We need to multiply the polynomial ( x − 4 ) by the polynomial ( x 2 − 3 x + 5 ) . This involves distributing each term of the first polynomial to each term of the second polynomial.

Multiplying by x First, we multiply x by each term in ( x 2 − 3 x + 5 ) : x ( x 2 − 3 x + 5 ) = x 3 − 3 x 2 + 5 x

Multiplying by -4 Next, we multiply − 4 by each term in ( x 2 − 3 x + 5 ) : − 4 ( x 2 − 3 x + 5 ) = − 4 x 2 + 12 x − 20

Adding the Polynomials Now, we add the two resulting polynomials: ( x 3 − 3 x 2 + 5 x ) + ( − 4 x 2 + 12 x − 20 ) Combining like terms, we have: x 3 + ( − 3 x 2 − 4 x 2 ) + ( 5 x + 12 x ) − 20 x 3 − 7 x 2 + 17 x − 20

Final Result Therefore, the product of ( x − 4 ) ( x 2 − 3 x + 5 ) is x 3 − 7 x 2 + 17 x − 20 .


Examples
Polynomial multiplication is used in various fields such as engineering, physics, and computer science. For example, in control systems, the transfer function of a system can be represented as a ratio of two polynomials. Multiplying these polynomials helps in analyzing the system's behavior and designing controllers. In computer graphics, polynomial multiplication is used in curve and surface modeling.

Answered by GinnyAnswer | 2025-07-04