Question

sari is factoring the polynomial $2x^2 + 5x + 3$. if one factor is $(x + 1)$, what is the other factor?
\bigcirc $2x - 3$
\bigcirc $2x + 3$
\bigcirc $3x - 2$
\bigcirc $3x + 2$

Explanation:

Step1: Recall polynomial division or factoring

To find the other factor, we can divide the polynomial \(2x^2 + 5x + 3\) by \((x + 1)\) or use factoring by grouping. Let's use polynomial long division.
Divide \(2x^2 + 5x + 3\) by \(x + 1\).
First term: \(\frac{2x^2}{x}=2x\). Multiply \((x + 1)\) by \(2x\): \(2x(x + 1)=2x^2 + 2x\).
Subtract from the original polynomial: \((2x^2 + 5x + 3)-(2x^2 + 2x)=3x + 3\).
Next term: \(\frac{3x}{x}=3\). Multiply \((x + 1)\) by \(3\): \(3(x + 1)=3x + 3\).
Subtract: \((3x + 3)-(3x + 3)=0\). So the quotient is \(2x + 3\).
Alternatively, we can use the fact that if \((x + 1)\) is a factor, then \(f(-1)=0\) (Factor Theorem), but we need the other factor. Let the other factor be \(ax + b\). Then \((x + 1)(ax + b)=ax^2+(a + b)x + b\). Comparing with \(2x^2 + 5x + 3\), we have \(a = 2\), \(b = 3\) (since \(b = 3\) and \(a + b = 5\) gives \(a = 2\)). So the other factor is \(2x + 3\).

Step2: Verify the factor

Multiply \((x + 1)\) and \(2x + 3\): \((x + 1)(2x + 3)=2x^2 + 3x + 2x + 3=2x^2 + 5x + 3\), which matches the original polynomial.

Answer:

\(2x + 3\) (corresponding to the option "2x + 3")