Problem 40 - A Graphical Approach to College Algebra

Question

Factor each perfect square trinomial completely. $$(5 r+2 s)^{2}+6(5 r+2 s)+9$$

Step-by-Step Solution

Answer

The expression factors as \((5r+2s+3)^2\).

Step 1: Identify the Form

Recognize the form of the trinomial \[(a + b)^2 + 2(a + b) + 1 \]which is similar to a perfect square trinomial represented as \[(x + y)^2 + 2(x + y) imes z + z^2\].

Step 2: Assign Values

Assign \( a = 5r + 2s \) and the constant term \( b = 3 \) from the expanded form of \((5r + 2s + 3)^2\). The constant term \( b^2 = 9 \), which implies \( b = 3 \).

Step 3: Rewrite the Expression

Now rewrite the whole expression:\[(5r + 2s + 3)^2\].Here, the middle term is double the product of initial parameter and constant: \(2 \times (5r + 2s) \times 3 = 6 \times (5r + 2s) \). Hence, it confirms the formation.

Step 4: Verification

Verify by expanding the squared term:\[(5r + 2s + 3)^2 = (5r + 2s)^2 + 2 \times (5r + 2s) \times 3 + 3^2\]This results in:\[(5r + 2s)^2 + 6(5r + 2s) + 9\]. Thus, the factorization is confirmed.

Key Concepts

Perfect Square Trinomial

Understanding perfect square trinomials is key when dealing with algebraic expressions that can be factored neatly. A perfect square trinomial is an expression that is the square of a binomial. Its general form is

• \((x + y)^2 = x^2 + 2xy + y^2\)

Here, the first term is the square of the first element of the binomial, the second term is twice the product of the two elements, and the third term is the square of the second element.
In the given problem, the trinomial looks very similar:

• \((5r + 2s)^2 + 6(5r + 2s) + 9\)

You can see how it fits the pattern of a perfect square trinomial by recognizing that

• \(5r + 2s\)

plays the role of a single variable in the formula.
Factorizing perfect square trinomials involves recognizing this structure and expressing the trinomial as the square of a binomial.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They follow rules for operations and simplifications just like the ones we use for numbers. In fact, understanding the rules for operating within algebraic expressions is crucial for solving equations and simplifying problems.

• The expression \((5r + 2s)^2 + 6(5r + 2s) + 9\) is an algebraic expression combining a square term, a multiply-and-add term, and a constant.

The given expression is reorganized into a simpler form, revealing its structure as a perfect square trinomial. Recognizing this enables an algebraic expression to be rewritten
or factored into

• \((5r + 2s + 3)^2\)

effectively." Each term of the expression, whether a product of variables and numbers or constants, contributes to the identity of the trinomial.

Quadratic Equations

Quadratic equations can appear in different forms in algebra. They are typically in the format of

• \(ax^2 + bx + c = 0\)

where

• \(a, b,\)
• and \(c\) are constants.

Comparatively, the original trinomial we are working on is

• \((5r + 2s)^2 + 6(5r + 2s) + 9\)

which resembles a quadratic equation expression. By manipulating such expression, it can be expressed in factored form as a binomial square.

• \((5r + 2s + 3)^2\).

Recognizing perfect square trinomials within quadratic equations helps in efficiently solving and simplifying these expressions.
Factoring helps in reducing complex algebraic terms into manageable components to identify solutions more effectively, especially useful in solving quadratic equations.