Problem 26 - Beginning and Intermediate Algebra

Question

Factor completely, if possible. Check your answer. $$z^{2}-10 z+24$$

Step-by-Step Solution

Answer

The completely factored expression is \( (z - 4)(z - 6) \).

Step 1: Identify the coefficients

In the given quadratic expression \(z^2 - 10z + 24\), the coefficient of the quadratic term is 1, the coefficient of the linear term is -10, and the constant term is 24.

Step 2: Find two numbers that multiply to give 24 and add to -10

We need to find two numbers that multiply to give 24 and add to give -10. Although there are many ways to find these numbers, one simple method is trial-and-error: - \(1 \times 24 = 24\), but \(1 + 24 ≠ -10\) - \((-1) \times (-24) = 24\), and \((-1) + (-24) = -25 ≠ -10\) - \(2 \times 12 = 24\), but \(2 + 12 ≠ -10\) - \((-2) \times (-12) = 24\), and \((-2) + (-12) = -14 ≠ -10\) - \(3 \times 8 = 24\), but \(3 + 8 ≠ -10\) - \((-3) \times (-8) = 24\), and \((-3) + (-8) = -11 ≠ -10\) - \(4 \times 6 = 24\), but \(4 + 6 ≠ -10\) - \((-4) \times (-6) = 24\), and \((-4) + (-6) = -10\) Hence, the two numbers are -4 and -6.

Step 3: Rewrite the expression as the product of two binomials

Since -4 and -6 are the numbers we're looking for, we can rewrite the given expression as the product of two binomials: \((z - 4)(z - 6)\)

Step 4: Check the answer

To check if our solution is correct, we can use the FOIL method to expand the product of binomials: \((z - 4)(z - 6) = z^2 - 6z - 4z + 24 = z^2 - 10z + 24\) As the expanded expression is the same as the given expression, our solution is correct. #Result#: The completely factored expression is \( (z - 4)(z - 6) \).

Key Concepts

Quadratic Equations

Quadratic equations are a powerful way to describe a variety of real-world problems. They are equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants.
The variable \( x \) represents an unknown that we need to solve.What makes quadratic equations unique is their highest exponent, which is 2. This gives the equation its parabolic shape when graphed.
The solutions to quadratic equations are the values of \( x \) where the equation equals zero.Typically, these solutions can be found via:

• Factoring
• Completing the square
• Using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Factoring is often the simplest method when the quadratic expression is factorable.
Understanding and being able to identify elements like the coefficients and constants play essential roles in solving these equations.

Factoring Methods

Factoring quadratic expressions involves rewriting them as a product of simpler terms.
This makes it easier to solve for the variable. Let's delve into the common factoring methods often used.Firstly, identifying numbers that add and multiply to specific values is crucial.
Let's consider the quadratic expression \( z^2 - 10z + 24 \). Here, the key task is choosing two numbers that multiply to give the constant term 24, while also adding up to the linear coefficient -10.
These numbers will help rewrite the quadratic expression as a product of two binomials.Trial-and-Error:

• We try combinations like \((-4)\) and \((-6)\) since \((-4) \times (-6) = 24\) and \((-4) + (-6) = -10\).

Rewriting and Verifying:

• Once these numbers are identified, rewrite the quadratic expression as \((z - 4)(z - 6)\).
• To verify, expand the binomials back to the original quadratic expression to ensure correctness.

This method of factoring allows for elegant solutions and often highlights the roots or zeros of the equation.

Polynomial Expressions

Polynomial expressions are mathematical expressions involving sums of powers in one or more variables. The term "polynomial" comes from "poly-" for many and "-nomial" meaning terms.
They can have any number of terms with non-negative integer exponents.In the context of quadratic expressions, they're a special kind of polynomial expression where the highest power of the variable is 2.
Polynomials like \( z^2 - 10z + 24 \) are standard forms in algebra and are crucial for understanding more complicated mathematical concepts.Properties of polynomials include being closed under addition, subtraction, and multiplication.
This means you can add, subtract, or multiply them without changing the polynomial's essential nature.

• Each individual term in a polynomial has a coefficient, which is the number in front of the variable.
• The highest degree term degree dictates the polynomial's degree. Here, \(z^2\) means it's a second-degree polynomial or quadratic.

Understanding polynomial operations, such as addition and multiplication, is foundational for manipulating expressions to simplify complex problems.
This enables analyzing and solving equations like quadratics by recognizing patterns and employing appropriate factoring methods.