Problem 3 - Elementary and Intermediate Algebra

Question

Fill in the blanks. Consider \(25 x^{2}+30 x+9\). a. The first term is the square of________. b. The last term is the square of ________. c. The middle term is twice the product of ________ and ________.

Step-by-Step Solution

Answer

a. \(5x\); b. 3; c. \(5x\) and 3.

Step 1: Identify the Expression Structure

Begin by recognizing that the expression \(25x^2 + 30x + 9\) resembles the form of a perfect square trinomial, which is \((ax + b)^2\), where:\( (ax)^2 = a^2x^2, \, 2abx, \, b^2 \).

Step 2: Analyze the First Term

The first term in the trinomial is \(25x^2\). This term is a square, which can be written as \((5x)^2\), where \(5x\) is the square root of \(25x^2\).

Step 3: Analyze the Last Term

The last term in the trinomial is 9. This term is a perfect square, which can be expressed as \(3^2\), where 3 is the square root of 9.

Step 4: Analyze the Middle Term

The middle term is \(30x\). According to the structure \(2abx\), it must equal \(2\times5x\times3\). Simplify \(2\times5\times3\) to confirm it equals 30. Hence, it represents twice the product of \(5x\) and 3.

Key Concepts

Squaring Binomials

When we talk about squaring a binomial, it involves multiplying a two-term algebraic expression by itself. If you have a binomial in the form of

• \((a + b)^2\)

this becomes \(a^2 + 2ab + b^2\) when you expand it. Each term comes from the identity of binomial squares.
Squaring a binomial results in a trinomial. The trick to grasp here is recognizing each part of the trinomial.

When you see a trinomial form like \(x^2 + 2xy + y^2\):

• The first term \(x^2\) is the square of the first binomial term \(x\).
• The last term \(y^2\) is the square of the second binomial term \(y\).
• The middle term \(2xy\) is twice the product of both terms from the binomial \(x\) and \(y\).

With this knowledge, you can quickly identify perfect square trinomials and their binomial roots!

Trinomial Expressions

Trinomials are algebraic expressions with three terms. In mathematics, recognizing specific patterns within trinomials can immensely help in simplifying and factoring them. The focus usually goes to special cases, such as perfect square trinomials.
Let's consider a trinomial expression

• \(ax^2 + bx + c\)

In some cases, it matches the form of a perfect square trinomial such asthe expression

• \( (mx + n)^2 \)

that expands to

• \(m^2x^2 + 2mnx + n^2\)

In identifying these perfect squares,

• first, look for a square in the first term, like \((5x)^2 = 25x^2\),
• then check the last term is a square like \(3^2 = 9\),
• and finally confirm the middle term, which is \(2 imes 5x imes 3 = 30x\).

Becoming familiar with these forms makes working with trinomials easier, and turns multisided puzzles into straightforward tasks!

Algebraic Identities

Algebraic identities are equations that are true for all values of the variables in the equation. These are basically the rules and formulas that govern algebraic operations. They act like shortcuts to help us quickly work with complex expressions. Consider a common identity with a perfect square trinomial:

• \((a + b)^2 = a^2 + 2ab + b^2\)

This formula helps you foresee the outcome of squaring any binomial. When you're faced with a trinomial like \(25x^2 + 30x + 9\), and you suspect it might be a perfect square, apply these identities:

• \(25x^2\) tells us \(a = 5x\),
• \(9\) tells us \(b = 3\), and
• \(30x\) confirms it by being \(2ab = 2 \times 5x \times 3\).

By recognizing these identities, you easily transform expressions and factor them with confidence. Mastery of algebraic identities turns algebra from a series of steps into a seamless dance of calculations.