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Chapter 5: Problem 201

Multiply the binomials. Use any method. $$ (2 x y+3)(3 x y+2) $$

Short Answer

Expert verified
6x^2y^2 + 13xy + 6

Step by step solution

01

- Apply the Distributive Property

Use the distributive property to multiply each term in the first binomial by each term in the second binomial. This means multiplying each term inside the parenthesis individually: If A = 2xy + 3 and B = 3xy + 2, then `(A)(B)` = `(2xy)(3xy) + (2xy)(2) + (3)(3xy) + (3)(2)`,
02

- Perform Multiplication for Each Term

Multiply each pair of terms: `2xy * 3xy = 6x^2y^2` `2xy * 2 = 4xy` `3 * 3xy = 9xy` `3 * 2 = 6`
03

- Combine Like Terms

Combine the like terms you obtained from multiplying each pair of terms: The terms `4xy` and `9xy` are like terms. Add them together: `6x^2y^2 + 4xy + 9xy + 6 = 6x^2y^2 + 13xy + 6`

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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 key to multiplying binomials and is used to simplify algebraic expressions. This property states that you need to multiply each term in the first binomial by every term in the second binomial. Essentially, you distribute each term across the other terms to ensure all possible combinations are covered. For example, let's take the binomials (2xy + 3) and (3xy + 2). According to the distributive property, you would multiply like this:
  • (2xy) * (3xy)
  • (2xy) * (2)
  • (3) * (3xy)
  • (3) * (2)
Breaking it down:
  • First, 2xy is distributed across 3xy and 2, resulting in 2xy * 3xy and 2xy * 2
  • Next, the term 3 is distributed across 3xy and 2, leading to 3 * 3xy and 3 * 2
By following this property, you ensure that all products are accounted for and easy to manage later on.
combining like terms
Combining like terms is an essential concept in algebra that simplifies expressions. Like terms are terms that have the same variable parts raised to the same power. In the expression obtained from multiplying the binomials, we have terms like 4xy and 9xy, which both contain the variable xy. To combine these like terms, you simply add their coefficients. In this example:
  • 4xy + 9xy = 13xy
Therefore, the expression 6x^2y^2 + 4xy + 9xy + 6 combines into 6x^2y^2 + 13xy + 6. Combining like terms makes it easier to work with algebraic expressions and helps to streamline equations.
intermediate algebra
Intermediate algebra builds on the basics by introducing more complex operations and concepts to solve equations and manipulate expressions. Key areas include:
  • Understanding polynomials: multiply and divide them, combine like terms, and distribute them.
  • Factoring: breaking down complex expressions into simpler factors.
  • Solving equations: finding variable values that satisfy the equation.
By mastering intermediate algebra, you gain the tools needed to solve more advanced problems and develop a deeper understanding of how algebra works. In multiplying binomials, you use multiple skills learned in intermediate algebra: the distributive property, recognizing and combining like terms, and understanding the structure of polynomials. This not only aids in simplifying expressions but also in solving larger, more complicated problems.

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Most popular questions from this chapter

Multiply the monomials. (a) \(\left(-10 x^{5}\right)\left(-3 x^{3}\right)\) (b) \(\left(\frac{5}{8} x^{3} y\right)\left(24 x^{5} y\right)\)

Multiply. Use either method. $$ (w-7)\left(w^{2}-9 w+10\right) $$

Multiply each pair of conjugates using the Product of Conjugates Pattern. $$ \left(12 p^{3}-11 q^{2}\right)\left(12 p^{3}+11 q^{2}\right) $$

For functions \(f(x)=7 x-8\) and \(g(x)=7 x+8\), find (a) \((f \cdot g)(x)\) (b) \((f \cdot g)(-2)\)

Square each binomial using the Binomial Squares Pattern. $$ \left(y+\frac{1}{4}\right)^{2} $$
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