Problem 198
Question
Factor. \(5 r^{2}+25 r+30\)
Step-by-Step Solution
Verified Answer
The polynomial factors to \(5(r + 2)(r + 3)\).
1Step 1 - 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.
2Step 2 - Factor out the GCF
Divide each term by the GCF (5) and factor it out. This gives:\[5(r^2 + 5r + 6)\]
3Step 3 - 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:\[r^2 + 5r + 6 = (r + 2)(r + 3)\]
4Step 4 - Combine the Factors
Combine the factors from Step 2 and Step 3 to get the final factored form:\[5(r + 2)(r + 3)\]
Key Concepts
greatest common factorquadratic expressionsfactoring process
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:
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.
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.
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:
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.
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.