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Chapter 0: Problem 32

\(y^{2}-4 y\)

Short Answer

Expert verified
The factored form of the given expression \(y^2 - 4y\) is \(y(y - 4)\).

Step by step solution

01

Identify the common factor of both terms

In the given expression \(y^2 - 4y\), we can see that both terms have 'y' as a common factor.
02

Factor out the common factor

Now, we can factor out the common term, 'y', from both terms. We obtain \[y(y - 4)\]
03

Write the final expression

After factoring, our simplified expression is: \[y(y - 4)\] So, the factored form of the given expression \(y^2 - 4y\) is \(y(y - 4)\).

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Key Concepts

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

Understanding Common Factors
When working with algebraic expressions like \(y^2 - 4y\), finding the common factor is the first crucial step in the factoring process. A common factor is a number or expression that divides each term in the equation without leaving a remainder. In our example, both terms \(y^2\) and \(-4y\) have the variable \(y\) in common. This means that \(y\) is a common factor.
To identify the common factor:
  • Look at the given expression's coefficients and variables.
  • Check for variables or numbers that appear in every term.
By recognizing the common factor \(y\), we keep the process of simplifying the expression straightforward and manageable.
Working with Polynomials
Polynomials are expressions consisting of variables and coefficients, combined using operations like addition, subtraction, and multiplication. In the expression \(y^2 - 4y\), we have a polynomial with two terms: \(y^2\) and \(-4y\). Each term is made up of variables raised to a power and a coefficient.
Understanding the structure of polynomials helps in factoring because:
  • It aids in identifying like terms and common factors.
  • It allows us to use algebraic properties efficiently for simplification.
Recognizing that the expression \(y^2 - 4y\) is a polynomial sets the stage to employ simplification techniques and arrive at its factored form.
Applying Simplification Techniques
Simplification techniques are essential in algebra to reduce expressions to their simplest form for easier problem-solving. In our polynomial \(y^2 - 4y\), after identifying \(y\) as the common factor, the next step involves factoring it out.
Here's how it's done:
  • Write the common factor \(y\) outside of a set of parentheses.
  • Inside the parentheses, divide each term of the polynomial by the common factor \(y\), resulting in \((y - 4)\).
This leaves us with the simplified expression \(y(y - 4)\). Such techniques help in breaking down complex polynomials into manageable pieces, simplifying calculations and providing clearer insights for further algebraic manipulations.

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Most popular questions from this chapter

By any method, determine all possible real solutions of each equation Check your answers by substitution. \(-x^{2}-2 x-1=0\)

Find all possible real solutions of each equation \(3 x^{6}-x^{4}-12 x^{2}+4=0\)

\((x+1)(x+2)^{2}+(x+1)^{2}(x+2)\)

\(20 y^{2}+7 y-3\)

\((x+y-x y)^{2}\)
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