Questions: Completely factor the given trinomial. 30 a^2 b-44 a b+16 b

Solution Steps

To factor the given trinomial \(30a^2b - 44ab + 16b\), we first observe that each term contains a common factor of \(b\). We can factor \(b\) out of the entire expression. After factoring out \(b\), we are left with a quadratic trinomial in terms of \(a\). We then look for two numbers that multiply to the product of the leading coefficient and the constant term, and add up to the middle coefficient. These numbers will help us factor the quadratic trinomial further.

The given trinomial is \(30a^2b - 44ab + 16b\). We first notice that each term contains a common factor of \(b\). We factor \(b\) out of the entire expression:

\[ b(30a^2 - 44a + 16) \]

Next, we focus on factoring the quadratic trinomial \(30a^2 - 44a + 16\). We need to find two numbers that multiply to \(30 \times 16 = 480\) and add up to \(-44\). These numbers are \(-24\) and \(-20\).

We can rewrite the middle term \(-44a\) as \(-24a - 20a\) and factor by grouping:

\[ 30a^2 - 24a - 20a + 16 = (30a^2 - 24a) + (-20a + 16) \]

Factor each group:

\[ = 6a(5a - 4) - 4(5a - 4) \]

Notice that \((5a - 4)\) is a common factor:

\[ = (6a - 4)(5a - 4) \]

Combine the factors with the common factor \(b\) that we factored out initially:

\[ b(6a - 4)(5a - 4) \]

Further simplify \(6a - 4\) by factoring out a 2:

\[ = 2b(3a - 2)(5a - 4) \]

Final Answer