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Chapter 6: Problem 96

Multiply the following. See Section 5.4 \(\left(y^{2}+3\right)\left(y^{2}-3\right)\)

Short Answer

Expert verified
The product is \(y^4 - 9\).

Step by step solution

01

Recognize the Pattern

Notice that the expression \(\left(y^{2}+3\right)\left(y^{2}-3\right)\) represents the difference of squares, which can be written as \(a^2 - b^2\) where \(a = y^2\) and \(b = 3\).
02

Apply the Difference of Squares Formula

Recall the difference of squares formula, which is \(a^2 - b^2 = (a+b)(a-b)\). In this case, \((y^2 + 3)(y^2 - 3) = (y^2)^2 - 3^2\).
03

Simplify the Expression

Apply the squares in the difference of squares: \((y^2)^2\) becomes \(y^4\) and \(3^2\) becomes \(9\). So, the expression is \(y^4 - 9\).

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Key Concepts

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

Polynomial Multiplication
Polynomial multiplication involves combining two or more algebraic expressions by multiplying them. This can often seem complicated at first, but specific methods can help simplify the process.
  • To multiply polynomials, apply the distributive property, which involves distributing each term in one polynomial to every term in the other polynomial.
  • A common pattern in polynomial multiplication is recognizing special products like the difference of squares.
  • In the given problem, instead of distributing each term individually, recognizing the difference of squares pattern greatly simplifies the multiplication.
Understanding how to multiply polynomials efficiently is crucial for simplifying complex algebraic expressions.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. In the context of the original exercise, the expression is made up of terms such as \(y^2\), constants like \(3\), and operations such as addition and subtraction.
  • Terms are the basic building blocks of algebraic expressions and can involve variables raised to an exponent.
  • Understanding how to work with variables and constants is essential, as manipulating these correctly is key to solving polynomial equations.
  • Algebraic expressions can represent a wide range of scenarios, from simple arithmetic operations to more complex equations like the difference of squares.
Recognizing patterns within these expressions, such as the difference of squares, is vital for simplifying and solving them efficiently.
Factoring in Algebra
Factoring is the process of breaking down a polynomial into simpler components, or factors, which, when multiplied together, give back the original polynomial. Factoring is an inverse operation to multiplication and is a fundamental skill in solving algebraic equations.
  • The difference of squares is a common factorization pattern. It states that the expression \(a^2 - b^2\) can be factored into \( (a + b)(a - b) \).
  • In the original exercise, recognizing the expression \((y^2 + 3)(y^2 - 3)\) as a difference of squares allows us to factor it into simpler terms quickly.
  • Mastering factoring techniques aids in simplifying complex expressions and solving polynomial equations efficiently.
Factoring not only simplifies algebraic expressions but also plays a crucial role in solving equations and finding roots.

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Most popular questions from this chapter

Factor each trinomial completely. See Examples I through II and Section 6.2. \(25 x^{2}-60 x y+36 y^{2}\)

Factor each trinomial completely. $$ x^{2}+\frac{1}{2} x+\frac{1}{16} $$

Factor each four-term polynomial by grouping. See Examples 11 through 16. $$ 6 x-42+x y-7 y $$

Solve each equation. See Examples I and 2. $$ (9 x+1)(4 x-3)=0 $$

Factor each binomial completely. \(27-t^{3}\)
See all solutions

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