Problem 97 Perform the operations. $$ 3... [FREE SOLUTION]

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Chapter 5: Problem 97

Perform the operations. $$ 3 y(y+2)+(y+1)(y-1) $$

Short Answer

Expert verified

The simplified expression is $4y^2 + 6y - 1$.

Step by step solution

Distribute the First Term

Start by distributing the term $3y$ into $y+2$. This gives us:\[3y(y+2) = 3y^2 + 6y\]

Expand the Second Term

Next, expand the term $(y+1)(y-1)$. Use the difference of squares identity which is $(a+b)(a-b) = a^2 - b^2$. Here, we get:\[(y+1)(y-1) = y^2 - 1^2 = y^2 - 1\]

Combine the Polynomials

Now, combine all the expanded terms from the first and second steps:\[3y^2 + 6y + y^2 - 1\]

Simplify by Combining Like Terms

Combine the like terms in the polynomial:1. Combine the $y^2$ terms: $3y^2 + y^2 = 4y^2$2. The $y$ terms remain the same: $+ 6y$3. The constant term is $-1$So, the simplified expression is:\[4y^2 + 6y - 1\]

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Key Concepts

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

Distribution in Algebra

In algebra, distribution is a powerful tool that simplifies expressions and makes problem-solving more straightforward. Distribution essentially involves multiplying a single term by each term inside a parenthesis.
In the exercise, we distribute the term $3y$ into $y+2$. This means we multiply $3y$ with $y$ and $3y$ with $2$. Let's break this down:

Multiply $3y$ by $y$: This gives $3y^2$, because $y \times y = y^2$.
Multiply $3y$ by $2$: This results in $6y$.

By understanding distribution, you simplify expressions systematically, making algebraic problems more manageable.
Practicing distribution with various terms hones problem-solving skills and ensures better handling of complex algebraic tasks.

Difference of Squares

The difference of squares is a specific algebraic pattern. This identity helps to quickly simplify expressions of the form $(a+b)(a-b)$. The resulting simplification is given by the formula: \[(a+b)(a-b) = a^2 - b^2\]In our exercise, we use it to simplify the product $(y+1)(y-1)$.
Here's how it works:

Think of $y+1$ as $a$ and $y-1$ as $b$.
Apply the identity: This simplifies to $y^2 - 1$, because $y^2 - 1^2 = y^2 - 1$.

This method helps to avoid lengthy multiplication and ensures calculations are more efficient.
Recognizing and using the difference of squares pattern is a key skill for tackling different polynomial products swiftly.

Combining Like Terms

Combining like terms is essential for simplifying expressions and achieving the final reduced form. Like terms are terms whose variables (and their exponents) are the same.
They can be added or subtracted to consolidate a polynomial.

Identify like terms: In $3y^2 + 6y + y^2 - 1$, the like terms are $3y^2$ and $y^2$.
Combine them: Adding these gives $4y^2$ (since $3y^2 + y^2 = 4y^2$).
The remaining terms are isolated: $+6y$ and $-1$, which have no similar terms, so they remain as is.

Through this process, we simplify the expression to $4y^2 + 6y - 1$.
Mastering the combination of like terms helps in reducing expressions to their simplest forms, making it easier to understand and work with complex algebraic expressions.

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91影视

Perform the operations. $$ 3 y(y+2)+(y+1)(y-1) $$

Short Answer

Step by step solution

Distribute the First Term

Expand the Second Term

Combine the Polynomials

Simplify by Combining Like Terms

Key Concepts

Distribution in Algebra

Difference of Squares

Combining Like Terms

One App. One Place for Learning.

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