Problem 254 Find each product. $$ (9 p+8... [FREE SOLUTION]

Chapter 5: Problem 254

Find each product. $$ (9 p+8 q)^{2} $$

Short Answer

Expert verified

81p^2 + 144pq + 64q^2

Step by step solution

Identify the Formula

Recognize that the expression $ (9p + 8q)^2 $ uses the square of a binomial formula, which is $ (a + b)^2 = a^2 + 2ab + b^2 $.

Apply the Formula

Substitute $ a = 9p $ and $ b = 8q $ into the formula: $(9p + 8q)^2 = (9p)^2 + 2(9p)(8q) + (8q)^2 $.

Calculate Each Term

Compute each term separately: $ (9p)^2 = 81p^2 \, \ 2(9p)(8q) = 144pq \, \ (8q)^2 = 64q^2 $.

Combine the Terms

Combine all the terms to get the final expression: $ 81p^2 + 144pq + 64q^2 $.

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Key Concepts

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

Square of a Binomial

The square of a binomial refers to multiplying a binomial by itself. A binomial is an algebraic expression that has exactly two terms, such as (9p + 8q).Binomial squaring follows a specific formula: \textbf{$(a + b)^{2} = a^{2} + 2ab + b^{2}$}.This formula lets you expand the product without directly multiplying the binomial twice.

For example, when squaring (9p + 8q), we recognize that we can use the formula by setting a = 9p and b = 8q.Then substitute these values back into the formula.

This makes it easier to manage and solve such problems.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and operators like addition and multiplication. For example, (9p + 8q) and 81p^2 + 144pq + 64q^2 are both algebraic expressions.

An essential aspect of working with algebraic expressions is understanding how to simplify and manipulate them. This often involves recognizing patterns and applying mathematical formulas, like the binomial square formula.

When substituting values or expressions to solve a problem, ensure that each step is clear and accurate. This helps in avoiding mistakes and quickly finding the correct solution.

Polynomial Multiplication

Polynomial multiplication extends the concepts of multiplying individual terms and combining them into a larger expression. A polynomial is an algebraic expression with more than two terms.

In our example, squaring a binomial (9p + 8q)^2 involves multiplying every term by itself and by every other term. Using the binomial formula, we break it down into smaller, manageable calculations:

Square the first term: (9p)^2 = 81p^2
Multiply both terms and double it: 2(9p)(8q) = 144pq
Square the second term: (8q)^2 = 64q^2

Combining these results gives us the final polynomial: 81p^2 + 144pq + 64q^2.

This process highlights the importance of breaking down complex problems into simpler steps, making polynomial multiplication more approachable.

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91影视

Find each product. $$ (9 p+8 q)^{2} $$

Short Answer

Step by step solution

Identify the Formula

Apply the Formula

Calculate Each Term

Combine the Terms

Key Concepts

Square of a Binomial

Algebraic Expressions

Polynomial Multiplication

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