To factor the quadratic polynomial [tex]\(3x^2 + 10x + 3\)[/tex], we need to find two binomials whose product equals the given polynomial. These binomials will be in the form [tex]\((ax + b)(cx + d)\)[/tex].
The polynomial [tex]\(3x^2 + 10x + 3\)[/tex] can be factored as follows:
1. Identify the coefficients: [tex]\(a = 3\)[/tex], [tex]\(b = 10\)[/tex], and [tex]\(c = 3\)[/tex].
2. We are looking for two numbers that multiply to [tex]\(3 \cdot 3 = 9\)[/tex] and add up to [tex]\(10\)[/tex].
3. The pair of numbers that satisfy these conditions are [tex]\(9\)[/tex] and [tex]\(1\)[/tex], because [tex]\(9 \cdot 1 = 9\)[/tex] and [tex]\(9 + 1 = 10\)[/tex].
4. Based on these numbers, we can rewrite the middle term [tex]\(10x\)[/tex] as [tex]\(9x + x\)[/tex]. This changes the polynomial to:
[tex]\[
3x^2 + 9x + x + 3
\][/tex]
5. Group the terms:
[tex]\[
(3x^2 + 9x) + (x + 3)
\][/tex]
6. Factor out the greatest common factor from each group:
[tex]\[
3x(x + 3) + 1(x + 3)
\][/tex]
7. Notice that [tex]\((x + 3)\)[/tex] is a common factor:
[tex]\[
(3x + 1)(x + 3)
\][/tex]
Thus, one factor of the polynomial [tex]\(3x^2 + 10x + 3\)[/tex] is [tex]\(x + 3\)[/tex].
