91Ó°ÊÓ

Quadratic expressions are polynomials of degree 2, which means the highest power of the variable is 2. They generally have the form:
\( ax^2 + bx + c \).
In this format:

• \(a\) is the coefficient of the quadratic term (\(x^2\))
• \(b\) is the coefficient of the linear term (\(x\))
• \(c\) is the constant term

Recognition of this form in any given expression is key before attempting further operations like factoring.
In our exercise, the quadratic expression is given as: \( z^2 - 3z + 28 \).
Here:

• \(a = 1\)
• \(b = -3\)
• \(c = 28\)

Factorization is the process of breaking down a polynomial into a product of simpler polynomials. For quadratic expressions, you generally look for two binomials that multiply to form the original quadratic expression.
The factorization is used to simplify polynomials and solve equations. To factor a quadratic expression of the form \(ax^2 + bx + c\), you look for two numbers that both:

• Multiply to the constant term \(c\)
• Add up to the linear coefficient \(b\)

If we apply this to our example, the constant term \(c\) is 28, and the linear coefficient \(b\) is -3. Once you list out the possible factor pairs of 28, you then check if any of those pairs can add up to -3.
For our expression, none of the pairs 1\*28, 2\*14, or 4\*7 add up to -3, indicating that our original quadratic expression \(z^2 - 3z + 28\) cannot be factored into integer components.

When factoring quadratic expressions, one useful technique is to list all the integer pairs that multiply to the given constant term. This step can help you quickly determine if the quadratic can be factored using integers.
For example, with the quadratic expression \( z^2 - 3z + 28 \), we need integer pairs that multiply to 28. The pairs are:

• 1 and 28
• 2 and 14
• 4 and 7

You then check them to see if any of these pairs add up to the linear coefficient (-3 in this case).
However, in this problem, none of the pairs of integers multiply to 28 while also summing to -3. This shows that not all quadratic expressions can be factored into integer coefficients. In such situations, other methods like the quadratic formula may be needed to solve the equation.