Factoring Trinomials
The given trinomial is \(3x^2 + 17x + 10\). To factor it we need to find two numbers that multiply to \(3x^2 \cdot 10\) (the coefficient of \(x^2\) times the constant term) and add up to \(17x\) (the coefficient of the middle term).
The two numbers that fit this criterion are 2 and 15, since:
\(2 \cdot 15 = 30\), and \(30\) is the product of the coefficient of \(x^2\) (3) and the constant term (10).
\(2 + 15 = 17\), which is the coefficient of the middle term.
Now we need to split the middle term, \(17x\), into two terms using the numbers 2 and 15:
\(3x^2 + 2x + 15x + 10\)
Next, we group terms:
\((3x^2 + 2x) + (15x + 10)\)
Factor out the common factor from each group:
\(x(3x + 2) + 5(3x + 2)\)
We now have a common binomial factor of \((3x + 2)\) which we can factor out:
\((3x + 2)(x + 5)\)
Therefore, the trinomial \(3x^2 + 17x + 10\) factors to \((3x + 2)(x + 5)\).
