The model represents a polynomial of the form [tex]$a x^2+$[/tex] [tex]$b x+c$[/tex]. Which equation is represented by the model?

A. [tex]$3 x^2-4 x-1=(3 x+1)(x-1)$[/tex]
B. [tex]$3 x^2-2 x-1=(3 x-1)(x+1)$[/tex]
C. [tex]$3 x^2-4 x+1=(3 x-1)(x-1)$[/tex]
D. [tex]$3 x^2-2 x+1=(3 x-1)(x-1)$[/tex]

Expand the right-hand side of each equation.
Compare the coefficients of x 2 , x , and the constant term on both sides of each equation.
Identify the equation where the coefficients match on both sides.
The correct equation is 3 x 2 − 4 x + 1 = ( 3 x − 1 ) ( x − 1 ) .

Explanation

Understanding the Problem We are given four equations and we need to determine which one is correct. The general form of the polynomial is a x 2 + b x + c . We will expand the right-hand side of each equation and compare the coefficients with the left-hand side.

Analyzing Equation 1 Let's analyze the first equation: 3 x 2 − 4 x − 1 = ( 3 x + 1 ) ( x − 1 ) . Expanding the right side, we get ( 3 x + 1 ) ( x − 1 ) = 3 x 2 − 3 x + x − 1 = 3 x 2 − 2 x − 1 . Comparing this with the left side, 3 x 2 − 4 x − 1 , we see that the coefficients of x are different ( − 4 vs. − 2 ). Thus, this equation is incorrect.

Analyzing Equation 2 Now let's analyze the second equation: 3 x 2 − 2 x − 1 = ( 3 x − 1 ) ( x + 1 ) . Expanding the right side, we get ( 3 x − 1 ) ( x + 1 ) = 3 x 2 + 3 x − x − 1 = 3 x 2 + 2 x − 1 . Comparing this with the left side, 3 x 2 − 2 x − 1 , we see that the coefficients of x are different ( − 2 vs. 2 ). Thus, this equation is incorrect.

Analyzing Equation 3 Let's analyze the third equation: 3 x 2 − 4 x + 1 = ( 3 x − 1 ) ( x − 1 ) . Expanding the right side, we get ( 3 x − 1 ) ( x − 1 ) = 3 x 2 − 3 x − x + 1 = 3 x 2 − 4 x + 1 . Comparing this with the left side, 3 x 2 − 4 x + 1 , we see that the coefficients of x 2 , x , and the constant term are the same on both sides. Thus, this equation is correct.

Analyzing Equation 4 Finally, let's analyze the fourth equation: 3 x 2 − 2 x + 1 = ( 3 x − 1 ) ( x − 1 ) . Expanding the right side, we get ( 3 x − 1 ) ( x − 1 ) = 3 x 2 − 3 x − x + 1 = 3 x 2 − 4 x + 1 . Comparing this with the left side, 3 x 2 − 2 x + 1 , we see that the coefficients of x are different ( − 2 vs. − 4 ). Thus, this equation is incorrect.

Conclusion Therefore, the correct equation is 3 x 2 − 4 x + 1 = ( 3 x − 1 ) ( x − 1 ) .


Examples
Understanding polynomial factorization is crucial in many areas, such as engineering, physics, and computer science. For example, when designing a bridge, engineers use polynomials to model the load distribution and ensure structural integrity. Factoring these polynomials helps them find critical points and understand the bridge's behavior under different conditions. Similarly, in computer graphics, polynomials are used to create curves and surfaces, and factoring them can optimize rendering processes and improve performance.

Answered by GinnyAnswer | 2025-07-05