Problem 77 - Beginning and Intermediate Algebra

Question

Factor completely by first taking out a negative common factor. $$-10 z^{2}+19 z-6$$

Step-by-Step Solution

Answer

\[-10z^2 + 19z - 6 = -1(10z - 3)(z - 2)\]

Step 1: Identify Common Factors

Here, we can see all terms in the polynomial have a common factor of -1. Now, let's factor out the -1.

Step 2: Factor out the Common Factor

Factor -1 from the given polynomial: \[-1(-10 z^{2}+19 z-6) = 10z^2 - 19z + 6\]

Step 3: Factor the Quadratic Expression

Now, we will factor the quadratic expression \(10z^2 - 19z + 6\). To do this, we can find two numbers that multiply to the product of the first and last terms (60) and add up to the middle term (-19). Let's use trial and error to find two such numbers: \[(10z - a)(z - b) = 10z^2 - 19z + 6\] We find that a = 3 and b = 2 satisfy the above conditions since: \[10z^2 - 3z - 2z + 6 = 10z^2 - 19z + 6\] So, we have factored the quadratic expression as follows: \[10z^2 - 19z + 6 = (10z - 3)(z - 2)\]

Step 4: Combine the Factored Terms

Now, let's combine the factored terms, including the -1 we factored out earlier: \[-1 (10 z^{2}+19 z-6) = -1 (10z - 3)(z - 2)\] #Final solution#: \[-10z^2 + 19z - 6 = -1(10z - 3)(z - 2)\]