Problem 55 Use the special product rules to... [FREE SOLUTION]

Chapter 5: Problem 55

Use the special product rules to find each product. $$[3 h+(k-1)]^{2}$$

Short Answer

Expert verified

9h^2 + 6hk - 6h + k^2 - 2k + 1

Step by step solution

Identify the Special Product Rule

The expression $ [3h + (k-1) ]^2 $ is a binomial squared. The rule for squaring a binomial is given by \[ (a + b)^2 = a^2 + 2ab + b^2 \]

Assign Variables

Assign $ a = 3h $ and $ b = k-1 $. Now rewrite the expression in terms of $ a $ and $ b $ as $ (a + b)^2 $.

Apply the Special Product Rule

According to the special product rule: \[ (3h + (k-1))^2 = (3h)^2 + 2(3h)(k-1) + (k-1)^2 \]

Simplify the Terms

Now simplify each term: \[ (3h)^2 = 9h^2 \] \[ 2(3h)(k-1) = 6h(k-1) = 6hk - 6h \] \[ (k-1)^2 = k^2 - 2k + 1 \]

Combine All Terms

Add all simplified terms together: \[ 9h^2 + 6hk - 6h + k^2 - 2k + 1 \]

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Key Concepts

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

headline of the respective core concept

The concept of a binomial squared comes in handy when dealing with polynomial expressions. A binomial is a mathematical expression containing two terms separated by a plus or minus sign. In this exercise, the given expression is \[ [3h + (k-1)]^2 \]. The 'squared' part signifies that we are multiplying the binomial by itself. Understanding the binomial squared rule helps in simplifying such expressions without having to individually multiply each term.

headline of the respective core concept

Simplifying algebraic expressions often involves breaking down complex expressions into simpler terms. For our binomial squared expression \[ [3h + (k-1)]^2 \], we start by recognizing it as \[ (a + b)^2 \]. Here, \[ a = 3h \] and \[ b = k-1 \]. Once identified, we apply the formula: \[ (a + b)^2 = a^2 + 2ab + b^2 \]. Each part of the formula needs simplification: \[ (3h)^2 = 9h^2 \], \[ 2(3h)(k-1) = 6hk - 6h \], and \[ (k-1)^2 = k^2 - 2k + 1 \]. Breaking down terms in this manner makes it easier to combine and simplify.

headline of the respective core concept

Applying algebraic rules is crucial for simplifying expressions. In this case, we use the binomial squared rule: \[ (a + b)^2 = a^2 + 2ab + b^2 \]. Identifying \[ a = 3h \] and \[ b = k-1 \] is the first step. Next, we substitute these values into the formula. Simplify each term: \[ 9h^2 \], \[ 6hk - 6h \], and \[ k^2 - 2k + 1 \]. Finally, combining all terms, gives us the simplified expression: \[ 9h^2 + 6hk - 6h + k^2 - 2k + 1 \]. Using these rules helps in handling similar problems efficiently.

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91影视

Use the special product rules to find each product. $$[3 h+(k-1)]^{2}$$

Short Answer

Step by step solution

Identify the Special Product Rule

Assign Variables

Apply the Special Product Rule

Simplify the Terms

Combine All Terms

Key Concepts

headline of the respective core concept

headline of the respective core concept

headline of the respective core concept

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