Problem 198 Factor. \(5 r^{2}+25 r+30\)... [FREE SOLUTION]

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Chapter 7: Problem 198

Factor. \(5 r^{2}+25 r+30\)

Short Answer

Expert verified

The polynomial factors to \(5(r + 2)(r + 3)\).

Step by step solution

- Identify the Greatest Common Factor (GCF)

Identify the greatest common factor of the terms in the polynomial. The terms are 5, 25, and 30. The GCF is 5.

- Factor out the GCF

Divide each term by the GCF (5) and factor it out. This gives:\[5(r^2 + 5r + 6)\]

- Factor the Quadratic Expression

Now, factor the quadratic expression \(r^2 + 5r + 6\). Find two numbers that multiply to 6 and add up to 5. These numbers are 2 and 3, so:\[r^2 + 5r + 6 = (r + 2)(r + 3)\]

- Combine the Factors

Combine the factors from Step 2 and Step 3 to get the final factored form:\[5(r + 2)(r + 3)\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

greatest common factor

To start with factoring polynomials like in our example, it's crucial to identify the Greatest Common Factor (GCF). The GCF is the largest number that divides all the coefficients of the polynomial terms. For the polynomial given, the terms are 5, 25, and 30. By finding their greatest common factor, we simplify our work.

Let's break this down:

5 can only be divided by 1 and 5
25 can be divided by 1, 5, and 25
30 can be divided by 1, 2, 3, 5, 6, 10, 15, and 30

Only the number 5 is common across all terms. Therefore, the GCF is 5. Factoring out the GCF can greatly simplify a polynomial, making the subsequent steps easier to manage.

quadratic expressions

Quadratic expressions are polynomials of the form to handle these, we often use factoring techniques.

In the expression from our example, we get: To factor this quadratic expression, we need to find two numbers that multiply to the constant term (6) and add up to the linear coefficient (5). By trial and intuition, we find that 2 and 3 fit these criteria. Hence:
Now, the quadratic expression has been successfully factored.

factoring process

The factoring process involves several key steps that simplify a polynomial into its constituent parts.
Let's walk through the process using our example:

First, identify the problem: Factoring
Step 1: Identify the GCF, which we found to be 5.

Step 2: Factor out the GCF: This gives us
Step 3: Factor the quadratic expression inside the parentheses, leading to
The quadratic expression breaks into
gives us the terms and that satisfy the conditions.

Step 4: Combine the factors from Steps 2 and 3. The final factored form is

Each step streamlines the polynomial, making it more understandable and easier to solve.
Practicing this process makes it second nature, simplifying complex problems into manageable steps.

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91影视

Factor. \(5 r^{2}+25 r+30\)

Short Answer

Step by step solution

- Identify the Greatest Common Factor (GCF)

- Factor out the GCF

- Factor the Quadratic Expression

- Combine the Factors

Key Concepts

greatest common factor

quadratic expressions

factoring process

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