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Chapter 6: Problem 10

Factor completely. Identify any prime polynomials. $$ 5 y^{2}+30 y+45 $$

Short Answer

Expert verified
The completely factored form of 5y^2 + 30y + 45 is 5(y + 3)^2.

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Identify the greatest common factor of the terms in the polynomial. The terms are 5, 30, and 45. The GCF of 5, 30, and 45 is 5.
02

Factor out the GCF

Factor out the GCF, which is 5, from each term in the polynomial: Step 2 Content: \[ 5(y^2 + 6y + 9) \].
03

Factor the Quadratic

We are left with a quadratic expression inside the parentheses: \(y^2 + 6y + 9\). Factor this quadratic by finding two numbers that multiply to 9 (the constant term) and add to 6 (the coefficient of the linear term). These numbers are 3 and 3. Therefore, we can write: \( y^2 + 6y + 9 = (y + 3)(y + 3) = (y + 3)^2 \).
04

Write the Final Factored Form

Combine the factored terms to write the completely factored form of the original polynomial: \[ 5(y + 3)^2 \].

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Key Concepts

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

The Greatest Common Factor (GCF)
To factor a polynomial, start by identifying the greatest common factor (GCF) among its terms. The GCF is the largest number that evenly divides all terms. For the polynomial 5y^{2} + 30y + 45 , we identify the coefficients of each term, which are 5, 30, and 45.
Finding their GCF involves listing the factors of each number:
  • 5: 1, 5
  • 30: 1, 2, 3, 5, 6, 10, 15, 30
  • 45: 1, 3, 5, 9, 15, 45
The common factor among these numbers is 5. This means that 5 is the greatest common factor for this set of numbers.
Quadratic Expression
After factoring out the GCF, you will often be left with a quadratic expression. A quadratic expression is a polynomial of the form ax^2 + bx + c . In our example, factoring out the GCF 5 from the polynomial 5y^{2} + 30y + 45 leaves us with y^2 + 6y + 9 inside parentheses.
To factor this quadratic expression, we look for two numbers that multiply to the constant term (9) and add to the coefficient of the linear term (6). Here, those numbers are both 3, giving us:
y^2 + 6y + 9 = (y + 3)(y + 3) = (y + 3)^2.
Completely Factored Form
The last step in factoring a polynomial is to express it in completely factored form. After factoring out the GCF and simplifying the quadratic expression, we combine all the factors. In our example: Starting with 5y^{2} + 30y + 45 , we factored out 5 to get:
5(y^2 + 6y + 9) Factoring the quadratic expression y^2 + 6y + 9 gave us (y + 3)^2.
Thus, the completely factored form of our original polynomial is:
5(y + 3)^2.

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