Problem 54 Factor completely. \(5 y^{2}-5... [FREE SOLUTION]

Chapter 5: Problem 54

Factor completely. \(5 y^{2}-5 y-30\)

Short Answer

Expert verified

5(y - 3)(y + 2)

Step by step solution

Identify the Greatest Common Factor (GCF)

First, determine the greatest common factor of the coefficients and the constants in the polynomial. In the expression 5测虏 - 5y - 30, 5 is the GCF.

Factor out the GCF

Factor 5 out of each term: 5(y虏 - y - 6)

Factor the Quadratic Expression

Next, factor the quadratic expression inside the parentheses. Look for two numbers that multiply to -6 (the constant term) and add to -1 (the coefficient of the linear term). These numbers are -3 and 2.

Write the Factored Form

Express the quadratic expression as a product of two binomials: 5(y - 3)(y + 2)

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Key Concepts

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

Greatest Common Factor (GCF)

To factor polynomials, finding the Greatest Common Factor (GCF) is often the first step. The GCF is the largest number or variable term that divides all the terms in the polynomial.

For the polynomial given in our example:

The terms are: 5测虏, -5y, and -30.
The GCF of the coefficients (5, -5, and -30) is 5.

So, you factor out the 5 from each term:

\[5(y虏 - y - 6)\]

This simplifies the polynomial and makes the next steps in factoring easier.

Quadratic Equations

Quadratic equations are polynomials of degree 2, typically written in the form ax虏 + bx + c.

In our example, after factoring out the GCF, we have a quadratic expression inside the parentheses: \(y虏 - y - 6\).

To factor a quadratic, we need to find two numbers that:

Multiply to the constant term \(-6\)
Add up to the coefficient of the middle term \(-1\)

In this case, the numbers are \(-3\) and \(2\). This is because:

\(-3 \times 2 = -6\)
\(-3 + 2 = -1\)

Therefore, we can express \(y虏 - y - 6\) as a product of two binomials:

\((y - 3)(y + 2)\)

Binomials

Binomials are algebraic expressions containing two terms which are separated by a plus or minus sign. In our example, the quadratic expression \(y虏 - y - 6\) was factored into two binomials: \((y - 3)\) and \((y + 2)\).

When we factor a quadratic expression, we look for binomials that, when multiplied together, give us the original quadratic polynomial.

In our final expression, we factored the polynomial completely:

\[5(y - 3)(y + 2)\]

This means our original polynomial \(5y^2 - 5y - 30\) is now represented as a product of a constant (the GCF) and two binomials. This form is much simpler and often helpful for solving equations or other algebraic manipulations.

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91影视

Factor completely. \(5 y^{2}-5 y-30\)

Short Answer

Step by step solution

Identify the Greatest Common Factor (GCF)

Factor out the GCF

Factor the Quadratic Expression

Write the Factored Form

Key Concepts

Greatest Common Factor (GCF)

Quadratic Equations

Binomials

One App. One Place for Learning.

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