Problem 65 - Algebra 1: Concepts and Skills

Question

Find the product. $$ (3 x+5)(3 x+5) $$

Step-by-Step Solution

Answer

The product of \( (3x+5)(3x+5) \) is \( 9x^2 + 30x + 25 \).

Step 1: Multiply the First Terms

The first terms of both binomials are '3x'. So, Multiplying first terms means \(3x * 3x = 9x^2\).

Step 2: Multiply the Outer Terms

The outer terms are '3x' and '5'. So, multiplied out they give \(3x * 5 = 15x\).

Step 3: Multiply the Inner Terms

The inner terms are '5' and '3x'. So, multiplied out they give \(5 * 3x = 15x\).

Step 4: Multiply the Last Terms

The last terms of both binomials are '5'. So, multiplied out they give \(5 * 5 = 25\).

Step 5: Combine the results

Adding all these gives the result \(9x^2 + 15x + 15x + 25 = 9x^2 + 30x + 25\).

Key Concepts

Binomial Multiplication

Binomial multiplication is a process in algebra that lets us find the product of two binomials. A binomial is simply an expression that has two terms. For example, the expression \((3x + 5)\) is a binomial because it contains two terms, '3x' and '5'.
When multiplying binomials, we use the distributive property to ensure each term in the first binomial is multiplied by each term in the second. An efficient way to remember this is with the acronym FOIL, which stands for:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outermost terms in the product.
• Inner: Multiply the middle terms.
• Last: Multiply the last terms in each binomial.

The final step after multiplying is to combine like terms, which means pairing and adding the terms that have the same variables raised to the same power.
In our example, multiplying \((3x + 5)(3x + 5)\) by following the FOIL method gives us:

1. First: \(3x \times 3x = 9x^2\)
2. Outer: \(3x \times 5 = 15x\)
3. Inner: \(5 \times 3x = 15x\)
4. Last: \(5 \times 5 = 25\)

Combine the terms to get the final product: \(9x^2 + 30x + 25\).
This process ensures that all combinations of terms are multiplied together, which is essential for finding products in algebra.

Quadratic Expressions

Quadratic expressions are a type of polynomial that play a crucial role in algebra. These expressions are characterized by having a degree of 2. In simpler terms, a quadratic expression will have its highest power as a square, such as \(x^2\).
The standard form of a quadratic expression is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) cannot be zero. Here, \(a\) is known as the leading coefficient, \(b\) as the linear coefficient, and \(c\) the constant term.
In our example, after multiplying the binomials \((3x + 5)(3x + 5)\), we derived a quadratic expression: \(9x^2 + 30x + 25\).
Understanding quadratic expressions is fundamental because they form the base of quadratic equations and functions. They can be used to model various real-world scenarios, from calculating projectile motions to maximizing areas in geometry.
To effectively work with quadratic expressions, it's important to be familiar with:

• Factoring Try finding numbers that multiply to the constant term and add to the linear coefficient.
• The Quadratic Formula: It's used to find the roots of quadratic equations and is written as \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Through practice, understanding how to manipulate and solve these expressions can greatly enhance problem-solving skills in algebra.

Polynomials

Polynomials are expressions that consist of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. In simple terms, a polynomial looks like this: \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\).
Each part of a polynomial separated by a plus or minus sign is called a "term". The degree of a polynomial is determined by the term with the highest degree (exponent).
In the example \(9x^2 + 30x + 25\), after multiplying the binomials, we have a quadratic polynomial of degree 2 since the highest exponent is 2.
Polynomials can be classified based on the number of terms, such as:

• Monomial: A single term (e.g., \(3x^2\)).
• Binomial: Two terms (e.g., \(x - 4\)).
• Trinomial: Three terms (e.g., \(x^2 + 5x + 6\)).

Polynomials are fundamental in algebra due to their widespread application. They're used in everything from calculating interest in finance to modeling curves in physics.
To work with polynomials effectively, it's crucial to become comfortable with their operations, including addition, subtraction, multiplication, division, and factoring. Remember, the order of terms in a polynomial matters; they're typically written from highest to lowest degree.