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Chapter 6: Problem 3

Are the following factored correctly? Check by multiplying. $$ y 2+2 y-8=(y+4)(y-2) $$

Short Answer

Expert verified
Yes, the expression is factored correctly.

Step by step solution

01

Multiply the Factors

To verify if the expression is factored correctly, we need to multiply the factors \((y + 4)(y - 2)\). We use the distributive property (also known as FOIL method for binomials) to expand this product.\[(y + 4)(y - 2) = y(y - 2) + 4(y - 2)\]
02

Apply the Distributive Property

Now, let's calculate each term:1. \(y(y - 2) = y^2 - 2y\)2. \(4(y - 2) = 4y - 8\)Thus:\[(y + 4)(y - 2) = y^2 - 2y + 4y - 8\]
03

Combine Like Terms

Combine the like terms \(-2y\) and \(4y\):\[y^2 - 2y + 4y - 8 = y^2 + 2y - 8\]
04

Compare With Original Expression

The expanded expression \(y^2 + 2y - 8\) matches the original expression \(y^2 + 2y - 8\). Therefore, the given factorization \((y + 4)(y - 2)\) is correct.

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Key Concepts

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

Distributive Property
The distributive property is a fundamental algebraic rule that is often used when dealing with expressions that need to be expanded or simplified. It states that \( a(b + c) = ab + ac \). This is true for any numbers \(a\), \(b\), and \(c\).
  • For expression multiplication, it helps distribute the term outside the parentheses to each term within.
  • In the exercise, we used the distributive property to expand \((y + 4)(y - 2)\) into two separate applications: first with \(y\) and then with \(4\).
To apply this property, multiply each term in one parenthesis by each term in the other. The distributive property is not just a multiplication tool鈥攊t鈥檚 foundational in simplifying expressions.
FOIL Method
The FOIL method is a specific application of the distributive property, tailored for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, describing the order in which you multiply the terms:
  • First: Multiply the first terms of each binomial.
  • Outer: Multiply the outermost terms in the product.
  • Inner: Multiply the inner terms.
  • Last: Multiply the last terms of each binomial.
For the expression \((y + 4)(y - 2)\), we executed:
  • First: \(y \times y = y^2\)
  • Outer: \(y \times (-2) = -2y\)
  • Inner: \(4 \times y = 4y\)
  • Last: \(4 \times (-2) = -8\)
Using the FOIL method is immensely helpful in keeping track of all terms when multiplying binomials.
Combining Like Terms
Combining like terms is a method used in algebra to simplify expressions or to solve equations. Like terms are terms that contain the same variables raised to the same power.
  • For example, in the expression \(y^2 - 2y + 4y - 8\), the terms \(-2y\) and \(4y\) are like terms because they both involve the variable \(y\) raised to the first power.
  • When you combine, or "add" these like terms, keep the variable part and add the coefficients: \(-2y + 4y = 2y\).
Combining like terms reduces the complexity of expressions, making further calculations or comparisons more straightforward. It鈥檚 an essential step especially when verifying or simplifying algebraic expressions.

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