Question

Multiply. $$(-h+9)(-h-9)$$

Short answer

Question: Multiply the given binomials and simplify the resulting expression: (-h + 9)(-h - 9). Answer: The simplified expression after multiplying the given binomials is $$h^2 - 81$$.

Step-by-step solution

Distribute the first term of the first binomial

We need to distribute the first term of the first binomial (-h) to both terms of the second binomial. This will give us: $$(-h)(-h) + (-h)(-9)$$

Distribute the second term of the first binomial

Now, we need to distribute the second term of the first binomial (9) to both terms of the second binomial. This will give us: $$9(-h) + 9(-9)$$

Combine the results of Step 1 and Step 2

Next, we will combine the results from Step 1 and Step 2 to get: $$(-h)(-h) + (-h)(-9) + 9(-h) + 9(-9)$$

Simplify the resulting expression

Finally, we will simplify each term of the expression and combine like terms: $$(-h)(-h) = h^2$$ $$(-h)(-9) = 9h$$ $$9(-h) = -9h$$ $$9(-9) = -81$$ So the simplified expression after combining like terms is: $$h^2 + 9h - 9h - 81$$ which simplifies to: $$h^2 - 81$$

Key concepts

Distributive Property

The distributive property is a fundamental concept in algebra. It lets you multiply a single term by each term inside a set of parentheses. This property is symbolized as \( a(b + c) = ab + ac \). It ensures that every term in one expression is multiplied by every term in another.
For example, in the exercise \((-h+9)(-h-9)\), we employ the distributive property to calculate the product of the two binomials. By distributing \(-h\) to each term in the second binomial, we get:

• \((-h)(-h)\)
• \((-h)(-9)\)

Then, we distribute the \(+9\) to each term in the second binomial:

• \(9(-h)\)
• \(9(-9)\)

By doing this, we ensure that we account for every possible product of terms between the two binomials. It’s like making sure to cross-multiply each term with its counterparts, systematically and thoroughly.
Mastering this process becomes essential in algebra as it helps simplify and solve expressions.

Like Terms

Like terms are essential in simplifying expressions. They are terms that have identical variables raised to the same power. For example, \(3h\) and \(-9h\) are like terms because both include the variable \(h\) raised to the first power. They can be added or subtracted due to their identical variable part, which allows simplification.
In the context of the exercise provided, once we've distributed the terms from both binomials, we ended up with \(9h\) and \(-9h\) in our expression. Recognizing these as like terms is critical. They can be combined, simplifying the expression further.
By adding \(9h\) and \(-9h\), we see that they cancel each other out to zero. Ensuring that like terms are properly identified and combined forms the backbone of simplifying algebraic expressions and solving equations.

Simplifying Expressions

Simplifying expressions is the process of making them as concise and straightforward as possible. This involves combining like terms and performing mathematical operations. Let's break down how simplification was used in our example step by step.
After applying the distributive property to \((-h+9)(-h-9)\), we ended up with:

• \(h^2\)
• \(+9h\) and \(-9h\)
• \(-81\)

Upon observing the expression: \(h^2 + 9h - 9h - 81\), it's apparent that \(9h\) and \(-9h\) are like terms and thus cancel each other out. This elimination streamlines our expression to \(h^2 - 81\).
The process of simplifying ensures that we are left with the most uncomplicated form of an expression. This simplified form is often easier to understand, work with, or use in further calculations, eliminating unnecessary complexity.