Problem 51 - College Algebra with Corequisite Support

Question

Brenda is placing tile on her bathroom floor. The area of the floor is \(15 x^{2}-8 x-7 \mathrm{ft}^{2}\). The area of one tile is \(x^{2}-2 x+1 \mathrm{ft}^{2}\). To find the number of tiles needed, simplify the rational expression: \(\frac{15 x^{2}-8 x-7}{x^{2}-2 x+1}\) Area \(=15 x^{2}-8 x-7\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{(3x + 1)(5x - 7)}{(x - 1)^2} \).

Step 1: Identify the Expression to Simplify

We need to simplify the rational expression given as \( \frac{15x^2 - 8x - 7}{x^2 - 2x + 1} \). This will help us find the number of tiles Brenda needs.

Step 2: Factor the Numerator

Factor the numerator of the expression, \( 15x^2 - 8x - 7 \). This quadratic can be factored as \( (3x + 1)(5x - 7) \).

Step 3: Factor the Denominator

Factor the denominator, \( x^2 - 2x + 1 \). Notice that it is a perfect square trinomial and can be factored as \( (x - 1)^2 \).

Step 4: Simplify the Expression

Now, substitute the factored forms back into the rational expression: \( \frac{(3x + 1)(5x - 7)}{(x - 1)^2} \). Since there are no common factors in the numerator and denominator, this is the simplified form of the expression.

Key Concepts

Factoring Quadratics

To simplify rational expressions, the first thing you might need to do is factor the quadratics involved. \(15x^2 - 8x - 7\) is an expression we want to simplify. This requires us to know how to factor quadratics effectively.

• First, identify the coefficient of \(x^2\), which is 15, the middle term's coefficient is -8, and the constant term is -7.
• We look for two numbers that multiply to give the product of the quadratic coefficient (15) and the constant (-7), i.e., -105 and add up to the middle term (-8).
• These numbers are -15 and 7. Rewrite the middle term using these numbers: \(15x^2 - 15x + 7x - 7\).
• Factor by grouping: \(15x(x - 1) + 7(x - 1)\).
• Factor out the common factor \((x-1)\): \((3x+1)(5x-7)\).

Now we have factored the quadratic \(15x^2 - 8x - 7\) into \((3x+1)(5x-7)\).

Simplifying Expressions

Simplifying expressions often means reducing them to their simplest form. For rational expressions, this means you need to factor both the numerator and the denominator and cancel out any common factors.

Consider the expression: \(\frac{15x^2 - 8x - 7}{x^2 - 2x + 1}\).

• The numerator \(15x^2 - 8x - 7\) is factored into \((3x+1)(5x-7)\).
• The denominator \(x^2 - 2x + 1\) is a perfect square trinomial, which simplifies into \((x-1)^2\).

After factoring, we plug back the factored forms into the rational expression: \(\frac{(3x + 1)(5x - 7)}{(x-1)^2}\).
The expression cannot be simplified further since no terms cancel out, so this is the simplest form.

Perfect Square Trinomials

A perfect square trinomial is a quadratic expression that can be rewritten as a square of a binomial. Recognizing these can be useful in simplifying rational expressions.

Consider the denominator of the given expression: \(x^2 - 2x + 1\). It is a perfect square trinomial.

• To recognize a perfect square trinomial, look at the first term's coefficient, the last term's coefficient, and check if the middle term is twice the product of the square roots of the first and last terms.
• Here, \(x^2\) is the square of \(x\) and \(1\) is the square of \(1\).
• The middle term \(-2x\) equals \(-2 \times x \times 1\), fitting the pattern.
• So, \(x^2 - 2x + 1 = (x-1)^2\).

Recognizing that \(x^2 - 2x + 1\) is \((x-1)^2\) helps us simplify expressions by identifying common structures.

Area Calculation

When calculating area, particularly with a variable involved, understanding how shapes, sizes, and expressions relate to each other is essential.

In Brenda's problem, she is tiling a bathroom floor. The goal is to determine how many tiles, each of a certain area, fit into a larger area.

• The total area of the floor is given by the quadratic expression \(15x^2 - 8x - 7\) square feet.
• The area of one tile is given by \(x^2 - 2x + 1\) square feet.
• To find out how many tiles are needed, you use the rational expression \(\frac{15x^2 - 8x - 7}{x^2 - 2x + 1}\), which represents the number of times the tile's area can fit into the floor's area.

After simplifying the rational expression, you understand how many tiles perfectly tile the area without overlap or gaps. Understanding the simplification process helps ensure an accurate tile count.