Problem 166 - Elementary Algebra

Question

Factor. \(5 y^{3}+30 y^{2}-35 y\)

Step-by-Step Solution

Answer

5y(y - 1)(y + 7)

Step 1: Identify the Greatest Common Factor (GCF)

Examine each term in the expression to find the greatest common factor. The terms in the expression are: - 5y^3 - 30y^2 - 35y The GCF of the coefficients (5, 30, and 35) is 5. All terms also have at least one factor of y. Therefore, the GCF is 5y.

Step 2: Factor out the GCF

Now factor out the GCF (5y) from each term in the expression: 5y( y^2 + 6y - 7 )

Step 3: Factor the Quadratic Expression

Next, factor the quadratic expression inside the parentheses: To factorize y^2 + 6y - 7, look for two numbers that multiply to -7 and add to 6. These numbers are 7 and -1.Rewrite the quadratic expression as follows: y^2 + 7y - y - 7 Group the terms: ( y^2 + 7y) - ( y + 7 ) Factor by grouping: y(y + 7) - 1(y + 7) Lastly, factor out the common binomial factor (y + 7): (y - 1)(y + 7) The complete factorization is: 5y(y + 7)( y - 1 )

Key Concepts

Greatest Common Factor

To start with, we need to determine the Greatest Common Factor (GCF) of the given polynomial expression. The GCF is the largest factor that divides all terms in the polynomial without leaving a remainder. Let's break this down:

• Analyze the coefficients of each term: 5, 30, and 35.
• The GCF of 5, 30, and 35 is 5 because 5 is the highest number that divides each of them exactly.
• Next, look at the variables. All terms (5y^3, 30y^2, and 35y) contain at least one 'y'. Hence, the GCF also includes 'y'.

So, the GCF for this polynomial is 5y. This means we can factor out a 5y from every term in the polynomial. This will simplify the polynomial and make the next steps easier to manage.

Quadratic Expressions

The next key concept involves working with quadratic expressions. A quadratic expression is any polynomial that fits the form ax^2 + bx + c. Here, we have to factorize the quadratic expression inside the parentheses after factoring out the GCF.

• After factoring out the GCF, we are left with: 5y(y^2 + 6y - 7).
• We need to factorize the quadratic expression y^2 + 6y - 7.
• To do this, we look for two numbers that multiply to -7 (the constant term) and add to 6 (the coefficient of the linear term).

In this case, the numbers are 7 and -1. Rewriting the quadratic expression, we get y^2 + 7y - y - 7, allowing us to move to the next step which is factorization by grouping.

Factoring by Grouping

Factoring by grouping is a method used to simplify polynomials by grouping terms with common factors. Here is a step-by-step guide to using this technique:

• Group the terms in pairs: (y^2 + 7y) and (-y - 7).
• Factor out the common factor in each group: y(y + 7) and -1(y + 7).
• Notice that (y + 7) is common in both groups. This allows us to factor it out, resulting in (y - 1)(y + 7).

The complete factorization process of the quadratic expression y^2 + 6y - 7 leads to (y - 1)(y + 7). Finally, including the GCF we factored out at the beginning, the fully factorized form of the original polynomial is 5y(y - 1)(y + 7). This method helps simplify and solve more complex polynomial equations efficiently.