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A common factor in algebra is a number or variable expression that divides every term in a polynomial without leaving a remainder. To find the common factor of multiple terms, it's important to identify the largest number that can divide all coefficients as well as any common variables at the same power level.

In the case of our trinomial, which is \(4z^{2} - 12z - 40\), we can spot that all the terms have a common numerical factor of 4. Here’s why:

• Each coefficient—4, 12, and 40—is divisible by 4.
• No variable is common across all terms, so only the numerical factor is significant here.

By factoring out the number, we simplify the expression to \(4(z^2 - 3z - 10)\). This step sets us up for easier handling of the polynomial.

Quadratic expressions are polynomials of degree two, generally in the form \(ax^2 + bx + c\). In our trinomial, the quadratic expression we are working with (after factoring out the common factor) is \(z^2 - 3z - 10\).

Quadratics can have up to two solutions. These are the values of \(z\) that will satisfy the expression when set to zero.

Understanding how to manage quadratic expressions is fundamental in algebra because of their frequent appearance in equations.

• They often describe parabolic graphs.
• When factoring, our goal is to express the quadratic as a product of binomials.

Factoring quadratics helps simplify solving them because it breaks down the expression into simpler components.

Factorization involves breaking down an algebraic expression into products of simpler sets of expressions, often binomials or constants. There are several techniques, but in our example, simple inspection or trial and error is effective.

Once we factored out the common factor, we were left with the quadratic \(z^2 - 3z - 10\). To factor it completely, we look for two numbers that simultaneously add to the middle coefficient \(-3\) and multiply to the constant term \(-10\).

• Here, these numbers are \(-5\) and \(+2\).
• These results give us the factors \((z - 5)\) and \((z + 2)\).

Finally, the complete factorization of the original trinomial is \(4(z - 5)(z + 2)\), combining both our use of common factors and quadratic factorization techniques. This form is more useful for solving equations or graphing, as each factor gives insight into the structure of the equation.