Problem 39 Multiply. See Example 7. $$ ... [FREE SOLUTION]

Chapter 5: Problem 39

Multiply. See Example 7. $$ (3 a+1)^{2} $$

Short Answer

Expert verified

The expanded form is $9a^2 + 6a + 1$.

Step by step solution

Understand the Problem

Our goal is to expand the expression $(3a + 1)^2$. This means we need to multiply the expression $(3a + 1)$ by itself.

Apply the Formula for Squaring a Binomial

Recall the formula $(x + y)^2 = x^2 + 2xy + y^2$. In this problem, $x = 3a$ and $y = 1$. Substitute these into the formula.

Calculate $x^2$

Computing $(3a)^2$ gives us $9a^2$ because $3a \times 3a = 9a^2$.

Calculate $2xy$

The term $2xy$ is $2 \times 3a \times 1$. Calculate this to get $6a$.

Calculate $y^2$

The term $y^2$ is simply $1^2$, which equals 1.

Combine All Terms

We now combine all the terms: $9a^2 + 6a + 1$ to form the expanded expression.

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Key Concepts

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

Squaring a Binomial

When we square a binomial, we mean multiplying it by itself. A binomial has two terms, like x + y. Squaring it involves applying the formula:

$(x + y)^2 = x^2 + 2xy + y^2$.

This formula is a helpful tool because it saves time and simplifies calculations. Let's look at the process:
If you have a binomial (3a + 1), squaring it means you multiply (3a + 1) by itself. This leads to three specific tasks:

Calculate the square of the first term $3a$, giving $9a^2$.
Calculate double the product of the two terms, $2 imes 3a imes 1 = 6a$, the middle term.
Finally, square the second term $1$ to get $1$.

These calculations combine to form the expanded version of the binomial.

Algebraic Expressions

An algebraic expression includes numbers, variables, and operations. When squaring a binomial like (3a + 1), we work with algebraic expressions. They are fundamental in algebra because they represent quantities in different forms.
Breaking down (3a + 1)^2 into understandable pieces helps us grasp the concept of expressions:

3a represents a variable a multiplied by a coefficient 3.
The constant 1 adds a fixed value to the expression.
Operators like addition or multiplication are used to combine these parts.

Using the binomial squared formula, we can methodically expand and simplify algebraic expressions. This shows the power of symbols in representing mathematical operations concisely, turning a compact expression into a series of calculations.

Polynomial Multiplication

Polynomial multiplication extends what we've done when squaring a binomial. A polynomial can have more terms, like (x^2 + 2x + 1). Multiplying polynomials involves distributing each term of one polynomial across every term of the other polynomial.
When squaring a binomial (3a + 1), we perform multiplication contained within polynomials:

Distribute 3a to both 3a and 1, calculating
$3a imes 3a = 9a^2$, and $3a imes 1 = 3a$.
Next, distribute 1 to both 3a and 1, resulting in $1 imes 3a = 3a$ and $1 imes 1 = 1$.

Finally, combining these results gives us 9a^2 + 6a + 1.
This process of multiplying each element highlights how multiplication of polynomials incorporates rules of distribution and simplification to handle more complex algebraic forms.

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91影视

Multiply. See Example 7. $$ (3 a+1)^{2} $$

Short Answer

Step by step solution

Understand the Problem

Apply the Formula for Squaring a Binomial

Calculate \(x^2\)

Calculate \(2xy\)

Calculate \(y^2\)

Combine All Terms

Key Concepts

Squaring a Binomial

Algebraic Expressions

Polynomial Multiplication

One App. One Place for Learning.

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