Problem 6 Write each expression without pa... [FREE SOLUTION]

91影视

Beginning Algebra: A Guided Approach

Rosemary Karr, Marilyn Massey, R. David Gustafson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 9: Problem 6

Write each expression without parentheses. $$(z+2)^{2}$$

Short Answer

Expert verified

The expression $(z + 2)^2$ without parentheses is $z^2 + 4z + 4$.

Step by step solution

Write down the expression

The given expression is $(z + 2)^2$

Expand the expression

Take $(z + 2)^2$ and expand it to $(z + 2) * (z + 2)$

Apply the rule for adding polynomial expressions

Use the formula $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = z$ and $b = 2$. Substituting a and b into the formula gives $z^2 + 2 * 2z + 2^2$.

Simplify the expression

Simplify the expression to $z^2 + 4z + 4$

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Key Concepts

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

Polynomial Expansion

Polynomial expansion is about breaking down an expression like $(z + 2)^2$ into simpler terms. This helps in making complex equations easier to handle.
To expand $(z + 2)^2$, we start by writing it as $(z + 2) \times (z + 2)$.
This step shows you're multiplying two binomials together.

When expanding, make sure to multiply every term in the first binomial by every term in the second binomial.
Each combination of terms needs to be included in the expansion.

This methodical approach helps unearth each part of the equation that contributed to its initial formation.

Simplifying Expressions

Simplifying expressions is about making an equation as straightforward as possible. It involves combining like terms to reduce complexity.
For $z^2 + 4z + 4$, you start by evaluating similar types of terms, like the powers of $z$.

Identify and combine all like terms to shrink the expression to its simplest form.
The expression $z^2$ stays as is because no other $z^2$ terms exist.
Terms like $4z$ and constants such as $4$ are evaluated separately.

The goal is to ensure each step progressively leads to a more manageable and comprehensible result.

Binomial Theorem

The Binomial Theorem is a powerful tool for expanding expressions raised to a power, like $(z + 2)^2$.
It provides a formula to quickly expand binomials without manual multiplication.
The formula is $(a + b)^n = a^n + \binom{n}{1} a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \ldots + b^n$.

For $(z + 2)^2$, both $a$ and $b$ are identified, where $a = z$ and $b = 2$.
The theorem helps structure the expansion by considering each possible combination of $a$ and $b$.

Using this provides a framework making expansion faster and less prone to errors.

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91影视

Write each expression without parentheses. $$(z+2)^{2}$$

Short Answer

Step by step solution

Write down the expression

Expand the expression

Apply the rule for adding polynomial expressions

Simplify the expression

Key Concepts

Polynomial Expansion

Simplifying Expressions

Binomial Theorem

One App. One Place for Learning.

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