Problem 56 - Algebra and Trigonometry

Question

Factor the expression completely. $$ r^{2}-6 r s+9 s^{2} $$

Step-by-Step Solution

Answer

The expression factors to \((r - 3s)^2\).

Step 1: Identify the Form

The given expression is a quadratic in the form \( ax^2 + bxy + cy^2 \), where \( x = r \), \( y = s \), \( a = 1 \), \( b = -6 \), and \( c = 9 \). We recognize it as a perfect square trinomial.

Step 2: Confirm Perfect Square Trinomial

A perfect square trinomial takes the form \((x - y)^2 = x^2 - 2xy + y^2\). Here, \(-6rs = -2(3s)r\) and \(9s^2 = (3s)^2\). This matches the pattern for a perfect square trinomial.

Step 3: Factor the Expression

Since the expression matches the pattern \(x^2 - 2xy + y^2\), we can factor it as \((r - 3s)^2\).

Step 4: Write the Final Answer

The completely factored form of the expression \(r^2 - 6rs + 9s^2\) is \((r - 3s)(r - 3s) = (r - 3s)^2\).

Key Concepts

Perfect Square Trinomials

A perfect square trinomial is a special type of quadratic expression. It can be expressed as the square of a binomial. A binomial is simply two terms connected by an addition or subtraction sign. When we expand

• \((x+y)^2\)
• \((x-y)^2\)

we get a trinomial. This means our expression will have three terms. Let’s see how this works:

• \((x+y)^2 = x^2 + 2xy + y^2\)
• \((x-y)^2 = x^2 - 2xy + y^2\)

When you notice a quadratic expression following one of these patterns, you have found a perfect square trinomial. It can then be easily factored into the square of a simple binomial. So, next time you face an expression like \(r^2 - 6rs + 9s^2\), try to recognize it as a perfect square trinomial. This makes the factoring process quicker and simpler. You just find the binomial whose square will give you the trinomial.

Factoring Expressions

Factoring expressions is about breaking down complex expressions into simpler factors. It’s like finding the pieces that multiply together to recreate the original expression. When dealing with quadratic expressions, factoring makes solving equations easier. Especially when expressions are in quadratic form, such as

• \(ax^2 + bx + c\)

The goal of factoring is to express it as a product of simpler binomials or polynomials. In our worked example of \(r^2 - 6rs + 9s^2\), recognizing the expression as a perfect square trinomial allowed for straightforward factoring. We expressed it as

• \((r-3s)^2\)

Factoring not only helps in simplifying expressions but also in solving quadratic equations by setting factors to zero and solving for the variable. It’s a powerful tool and a major stepping stone in algebra.

Quadratic Expressions

Quadratic expressions are algebraic expressions of degree 2. The general form is presented as \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Quadratics are ubiquitous in mathematics and science because they describe phenomena like projectile motion and parabolic shapes. Identifying a quadratic involves looking for the square term, typically

• like \(x^2\)

and ensuring the degree of no other term exceeds 2. Within the realm of quadratics, perfect square trinomials and their special patterns simplify many problems. When you see a quadratic expression:

• Check if it matches a familiar pattern
• Like the perfect square trinomial or another factorable form

With \(r^2 - 6rs + 9s^2\), recognizing the structure allowed us to factor it into a simple, neat binomial square. Understanding quadratic expressions is critical. It lays a foundational understanding for solving numerous mathematical problems in algebra.