Simplify the expression $5(x-3)(x^2+4x+1)$.

The expression 5 ( x − 3 ) ( x 2 + 4 x + 1 ) simplifies to 5 x 3 + 5 x 2 − 55 x − 15 .
;

Answered by Anonymous | 2025-07-04

Expand the product of the two polynomials: ( x − 3 ) ( x 2 + 4 x + 1 ) = x 3 + x 2 − 11 x − 3 .
Multiply the result by the constant 5: 5 ( x 3 + x 2 − 11 x − 3 ) = 5 x 3 + 5 x 2 − 55 x − 15 .
Compare the simplified expression with the given multiple-choice options.
The simplified expression is 5 x 3 + 5 x 2 − 55 x − 15 ​ .

Explanation

Understanding the Problem We are asked to simplify the expression 5 ( x − 3 ) ( x 2 + 4 x + 1 ) . This involves expanding the product of a constant and two polynomials.

Expanding the Polynomials First, we expand the product of the two polynomials ( x − 3 ) ( x 2 + 4 x + 1 ) . We can do this by multiplying each term in the first polynomial by each term in the second polynomial:


( x − 3 ) ( x 2 + 4 x + 1 ) = x ( x 2 + 4 x + 1 ) − 3 ( x 2 + 4 x + 1 )

Distributing the Terms Now, we distribute x and − 3 into the second polynomial:

x ( x 2 + 4 x + 1 ) = x 3 + 4 x 2 + x
− 3 ( x 2 + 4 x + 1 ) = − 3 x 2 − 12 x − 3

Combining Like Terms Next, we combine these two results:

( x 3 + 4 x 2 + x ) + ( − 3 x 2 − 12 x − 3 ) = x 3 + ( 4 x 2 − 3 x 2 ) + ( x − 12 x ) − 3 = x 3 + x 2 − 11 x − 3

Multiplying by the Constant Now, we multiply the simplified polynomial by the constant 5:

5 ( x 3 + x 2 − 11 x − 3 ) = 5 x 3 + 5 x 2 − 55 x − 15

Final Answer Comparing our simplified expression 5 x 3 + 5 x 2 − 55 x − 15 with the given multiple-choice options, we see that it matches the fourth option.

Therefore, the simplified expression is 5 x 3 + 5 x 2 − 55 x − 15 .
Examples
Polynomial simplification is a fundamental skill in algebra and is used in various real-world applications. For example, engineers use polynomials to model curves and surfaces in computer-aided design (CAD). Simplifying polynomial expressions allows them to optimize designs, reduce material usage, and improve performance. In economics, polynomials can represent cost and revenue functions, and simplifying these expressions helps businesses analyze profitability and make informed decisions about pricing and production levels. Understanding polynomial simplification provides a foundation for solving complex problems in science, engineering, and economics.

Answered by GinnyAnswer | 2025-07-04