Answered

Sari is factoring the polynomial $2 x^2+5 x+3$. If one factor is ($x+1$), what is the other factor?


A$2 x-3$
B$2 x+3$
C$3 x-2$
D$3 x+2$

Verified Answer :

  • Identify the structure of the other factor as \((Ax+B)\) such that \((x+1)(Ax+B) = 2x^2 + 5x + 3\).
  • Equate coefficients to find \(A=2\) and \(B=3\).
  • Verify the expansion \((x+1)(2x+3) = 2x^2+5x+3\).
  • The other factor is \(\boxed{2x + 3}\).

Explanation

  1. Analysis of the Problem

We are given a quadratic polynomial \(2x^2 + 5x + 3\) and one of its factors, which is \((x + 1)\). Our goal is to determine the other factor. We can denote the second factor as a linear expression of the form \((Ax + B)\), such that their product equals the original polynomial: \((x + 1)(Ax + B) = 2x^2 + 5x + 3\).

  1. Determining the Other Factor

By expanding the product \((x + 1)(Ax + B)\), we get:
\[(x + 1)(Ax + B) = Ax^2 + Bx + Ax + B = Ax^2 + (A + B)x + B\]
To match this with the given polynomial \(2x^2 + 5x + 3\), we compare the coefficients:

  1. The coefficient of \(x^2\) gives \(A = 2\).
  2. The constant term gives \(B = 3\).
  3. The coefficient of \(x\) gives \(A + B = 2 + 3 = 5\), which is consistent with the middle term of the original polynomial.

Thus, the other factor is \((2x + 3)\).

  1. Verification and Conclusion

We have successfully identified the other factor. Checking the expansion \((x + 1)(2x + 3) = 2x^2 + 3x + 2x + 3 = 2x^2 + 5x + 3\), we see that it perfectly matches the original polynomial. The correct option is the second one: \((2x + 3)\).

Examples

Factoring polynomials is a fundamental skill in algebra used to simplify complex equations, solve for roots, and understand the behavior of functions. For instance, if you are designing a rectangular garden and know its area is represented by \(2x^2 + 5x + 3\) square meters, factoring this expression helps you determine that the dimensions of the garden are \((x + 1)\) meters and \((2x + 3)\) meters. This makes it much easier to plan the layout and calculate the perimeter or other features of the garden!