Problem 44 - Beginning and Intermediate Algebra

Question

Factor completely by first taking out \(-1\) and then by factoring the trinomial, if possible. Check your answer. $$-z^{2}+4 z-4$$

Step-by-Step Solution

Answer

The short version of the answer is: \(-z^2 + 4z - 4 = -1(z - 2)(z - 2)\)

Step 1: Factor out the common factor of -1

We'll begin by factoring out the common factor of -1 from the given expression: \(-z^2 + 4z - 4 = -1(z^2 - 4z + 4)\)

Step 2: Factor the trinomial expression

Next, we will factor the quadratic (trinomial) expression inside the parentheses: \(z^2 - 4z + 4\) We are looking for two numbers that multiply to 4 (1*4) and add to -4. The numbers that meet these conditions are -2 and -2. \(z^2 - 4z + 4 = (z - 2)(z - 2)\)

Step 3: Put it all together

Now, we can combine the factoring we've done so far to express the original expression in its completed factored form: \(-z^2 + 4z - 4 = -1(z^2 - 4z + 4) = -1(z - 2)(z - 2)\)

Step 4: Check your answer

To check our answer, we will expand our factored expression and see if it matches the original expression: \(-1(z - 2)(z - 2) = -1(z^2 - 4z + 4) = -z^2 + 4z - 4\) Since our expanded expression matches the original expression, our factoring is correct. The final factored expression is: \(-z^2 + 4z - 4 = -1(z - 2)(z - 2)\)

Key Concepts

Trinomial Factoring

Trinomial factoring is a powerful method to simplify quadratic expressions by breaking them into the product of simpler binomials. It involves identifying expressions of the form \( ax^2 + bx + c \) and factoring them when possible. This requires using two numbers that multiply to \( ac \) (the product of the coefficient of \( x^2 \) and the constant term) and add to \( b \) (the coefficient of \( x \)). By focusing on factoring \( z^2 - 4z + 4 \), we observe it in the pattern of \( a^2 - 2ab + b^2 = (a-b)^2 \). Here, the trinomial can be recognized as a perfect square, \( (z-2)^2 \), because both \(-2\) and \(-2\) multiply to \(4\) and add to \(-4\). This makes the expression easy to factor and understand.

• Identify a perfect square trinomial: \( (z-2)(z-2) \).
• Recognize and factor regularly occurring squares.
• Practice makes this approach more intuitive.

Polynomial Expressions

Polynomial expressions are mathematical phrases that can involve variable powers, coefficients, and constants. In their simplest form, they are composed of terms such as \\[ 3x, -5x^2, \frac{1}{2}x \quad \text{or} \quad 7 \]. \These terms are combined using operations of addition, subtraction, and multiplication.These expressions, like \(-z^2 + 4z - 4\), are classified based on their highest power or degree. Quadratic polynomials, characterized by their degree of 2, are also known as second-degree polynomials. Understanding polynomials involves:

• Recognizing terms and their coefficients.
• Combining and simplifying like terms.
• Applying distributive properties when necessary.

Factoring polynomial expressions can often simplify problem-solving. Finding common factors, in this case \(-1\), allows us to reorganize and reduce complexity.

Quadratic Equations

Quadratic equations are equations that take the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). They can often be solved by factoring, like in the problem where we dealt with \(-z^2 + 4z - 4\). Solving these equations aims to find the values of \( x \) that satisfy the equation, known as the roots.Through factoring the quadratic expression \( z^2 - 4z + 4 \) to \( (z - 2)(z - 2) \), we identify the repeated root \( z = 2 \). The zero-product property then confirms solutions by setting each factor equal to zero and solving for \( x \):

• Transform expressions to standard form, if needed.
• Identify and factor the trinomial, when possible.
• Apply the zero-product property to solve for roots.

Understanding quadratic equations deepens comprehension of various algebraic structures, and solving them builds strong problem-solving skills.