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Chapter 12: Problem 67

Review factoring polynomials (Sections \(6.1-6.6)\) Factor $$ 6 y^{2}+36 y-240 $$

Short Answer

Expert verified
The factored form is \( 6(y + 10)(y - 4) \).

Step by step solution

01

Find the greatest common factor (GCF)

Identify the greatest common factor of the coefficients 6, 36, and -240. The GCF is 6.
02

Factor out the GCF

Divide each term by the GCF and factor it out. This gives: \[ 6(y^{2} + 6y - 40) \]
03

Factor the quadratic expression

Now factor the quadratic expression inside the parentheses. We need two numbers that multiply to -40 and add up to 6. These numbers are 10 and -4. Hence, the quadratic expression can be factored as: \[ y^2 + 6y - 40 = (y + 10)(y - 4) \]
04

Combine the factors

Combine the GCF and the factored quadratic expression to get the final factored form: \[ 6(y + 10)(y - 4) \]

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Key Concepts

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

Greatest Common Factor (GCF)
Understanding the Greatest Common Factor (GCF) is key in the first step of factoring polynomials. The GCF of a polynomial is the largest number that divides all the coefficients of the terms in the polynomial without leaving any remainder.

To find the GCF of 6, 36, and -240:
  • Break each number down into its prime factors.
  • 6 = 2 脳 3
  • 36 = 2^2 脳 3^2
  • -240 = -1 脳 2^4 脳 3 脳 5
Identify the common factors in all terms and select the lowest power of these common factors. The common factors here are 2 and 3, and the lowest powers are 2^1 and 3^1, respectively. So, 2 脳 3 = 6, making 6 the GCF.

By factoring out the GCF, you simplify the polynomial, making it easier to handle the remaining quadratic or other higher-order expressions.
Quadratic Expressions
Quadratic expressions are polynomials of degree 2. They have the standard form: \(ax^2 + bx + c\). These expressions can often be factored into the product of two binomials.

In the exercise, the polynomial inside the parentheses after factoring out the GCF is \(y^2 + 6y - 40\).

To factor this quadratic expression, you need to find two numbers that multiply to the constant term (in this case, -40) and add up to the coefficient of the middle term (in this case, 6). The numbers 10 and -4 satisfy these conditions because:
  • 10 脳 (-4) = -40
  • 10 + (-4) = 6
Therefore, the quadratic expression \(y^2 + 6y - 40\) can be factored as:\
\((y + 10)(y - 4)\).
Factoring Techniques
Factoring techniques vary depending on the type of polynomial and its degree. For quadratic expressions, several common methods include:

  • Factoring by grouping: Used mainly when there are four terms. Rearrange and group terms to factor out common binomials.
  • Factoring trinomials: Look for two numbers that multiply to the constant term and add to the coefficient of the linear term, as in the exercise.
  • Difference of squares: Identifying terms that are perfect squares and then writing them as the product of a binomial and its conjugate.
The exercise utilized the factoring trinomial technique to factor \(y^2 + 6y - 40\) into \((y + 10)(y - 4)\).

Combining the GCF and the factors of the quadratic expression gives the final factored form: \(6(y + 10)(y - 4)\). Remember, always check for a GCF before applying other factoring techniques, as it simplifies the entire process!

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