Problem 51 - College Algebra

Question

Brenda is placing tile on her bathroom fl or. The area of the fl or 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 fi \(\mathrm{d}\) the number of tiles needed, simplify the rational expression: \(\frac{15 x^{2}-8 x-7}{x^{2}-2 x+1}\).

Step-by-Step Solution

Answer

Brenda needs \(\frac{15x + 7}{x-1}\) tiles, with \(x \neq 1\).

Step 1: Factor the Numerator

The numerator of the expression is \(15x^2 - 8x - 7\). To simplify, we need to factor this quadratic equation. We find two numbers: one that multiplies to \(15 \times -7 = -105\) and adds to \(-8\). These numbers are \(7\) and \(-15\).Rewriting the expression: \(15x^2 + 7x - 15x - 7\). Group terms and factor them:\[15x^2 + 7x - 15x - 7 = (15x^2 + 7x) + (-15x - 7)\]Factor each group:\[= x(15x + 7) - 1(15x + 7)\]Combine the factors:\[= (x - 1)(15x + 7)\]

Step 2: Factor the Denominator

The denominator of the expression is \(x^2 - 2x + 1\). This is a perfect square trinomial which can be rewritten as:\[(x - 1)^2\]

Step 3: Simplify the Rational Expression

The rational expression we have is:\[\frac{15x^2 - 8x - 7}{x^2 - 2x + 1} = \frac{(x - 1)(15x + 7)}{(x - 1)(x - 1)}\]Cancel out the common factor \((x - 1)\) from the numerator and denominator:\[= \frac{15x + 7}{x - 1}\]

Step 4: Conclusion

The simplified rational expression \(\frac{15x + 7}{x - 1}\) represents the number of tiles Brenda needs for her bathroom floor. However, it is crucial to note that \(x eq 1\) because it would make the denominator zero, which is undefined in mathematics.

Key Concepts

Factoring Quadratics

Factoring quadratics is a key skill when working with polynomials. The process entails breaking down a quadratic expression, expressed as \(ax^2 + bx + c\), into a product of simpler polynomials.To factor a quadratic equation, you need two numbers that both sum up to the coefficient \(b\) and multiply to the product of \(a\) and \(c\).
In this case, the expression \(15x^2 - 8x - 7\) was factored by finding two numbers that multiply to \(-105\) (\(15 \times -7\)) and add to \(-8\).
These numbers are 7 and -15.
The expression \(15x^2 - 8x - 7\) can be rewritten and grouped as \(15x^2 + 7x - 15x - 7\), then factored into \((x-1)(15x+7)\).
This factorization allows the expression to be simplified more easily in later steps.

Simplifying Expressions

Simplifying expressions is all about reducing an expression to its simplest form.In rational expressions, this involves canceling out common factors in the numerator and denominator.
Once the quadratic on the numerator \(15x^2 - 8x - 7\) was factored into \((x-1)(15x+7)\), and the denominator \(x^2 - 2x + 1\) was recognized as a perfect square trinomial \((x-1)^2\), the expression could be simplified.
The rational expression \(\frac{(x-1)(15x+7)}{(x-1)(x-1)}\) allows the common factor \((x-1)\) to be canceled out in both the numerator and the denominator.
It's important to ensure the expression is not undefined, hence the value that makes the denominator zero should be identified and excluded.

Perfect Square Trinomials

Perfect square trinomials have a special form and are a type of quadratic expressions.They generally look like \(a^2 - 2ab + b^2\) or \(a^2 + 2ab + b^2\).
When factoring such trinomials, they tend to break down into \((a-b)^2\) or \((a+b)^2\) respectively.A perfect square trinomial is convenient because it can be simplified to a single binomial squared.
In the given exercise, the denominator \(x^2 - 2x + 1\) is a perfect square trinomial that simplifies to \((x-1)^2\).
Recognizing perfect square trinomials is crucial as it greatly simplifies the factoring process, ensuring easier simplification of any rational expression.