Problem 82 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 2 z^{2}+40 z+200 $$

Step-by-Step Solution

Answer

The factored form is 2(z+10)². There are no prime polynomials in the factored form.

Step 1: Identify the Greatest Common Factor (GCF)

Look for the greatest common factor in all the terms of the polynomial. The terms are 2z², 40z, and 200. The GCF of these terms is 2.

Step 2: Factor out the GCF

Factor out the GCF (2) from the polynomial. This gives: 2(z² + 20z + 100)

Step 3: Factor the Quadratic Expression

Now, focus on factoring the quadratic expression inside the parentheses: z² + 20z + 100. We look for two numbers that multiply to 100 and add to 20. These numbers are 10 and 10. So, we can write: z² + 20z + 100 = (z + 10)(z + 10) ...

Step 4: Write the Final Factored Form

Combine the GCF factor (2) with the factored quadratic expression to write the fully factored form: 2(z + 10)(z + 10). Simplifying, we get: 2(z + 10)².

Step 5: Identify Prime Polynomials

Check if any of the factors are prime polynomials. In this case, (z + 10) is not prime because it is a linear factor. Hence, there are no prime polynomials in the factored form.

Key Concepts

greatest common factor

The greatest common factor (GCF) is the highest number that can evenly divide all terms in a polynomial. It's an essential step in factoring because it simplifies the expression, making it easier to work with. For instance, in our example, we have the terms: \(2z^2\), \(40z\), and \(200\). The GCF of these terms is \(2\). By factoring out the GCF, we reduce the original polynomial to a simpler form: \(2(z^2 + 20z + 100)\). This simplification is crucial for the next steps in the factoring process.

quadratic expression

A quadratic expression is a polynomial of degree 2, which means the highest exponent is 2. It generally takes the form of \(ax^2 + bx + c\). In our example, once we've factored out the GCF, we are left with the quadratic expression \(z^2 + 20z + 100\). Factoring a quadratic expression often involves finding two numbers that multiply to the constant term (in this case, 100) and add to the linear coefficient (in this case, 20). Here, these numbers are both 10, so we can factor the quadratic as \((z + 10)(z + 10)\).

factoring techniques

Factoring techniques are methods used to break down polynomials into simpler components, which can be multiplied to obtain the original polynomial. Some common methods include:

• Factoring out the greatest common factor (GCF)
• Factoring by grouping
• Factoring trinomials
• Factoring perfect square trinomials
• Difference of squares

In our example, the quadratic expression \(z^2 + 20z + 100\) is a perfect square trinomial, which factors into \((z + 10)(z + 10)\). Combining this with the GCF factored out earlier, we get the fully factored form: \(2(z + 10)^2\).

prime polynomials

Prime polynomials are polynomials that cannot be factored further over the set of integers. They are analogous to prime numbers in arithmetic. In our example, the factor \(z + 10\) is a linear polynomial and not prime because it can be expressed in simpler linear terms. Thus, the final answer \(2(z + 10)^2\) indicates that there are no prime polynomials within this factorization. Always check each factor after factorization to see if any cannot be broken down further to ensure correct identification of prime polynomials.