Problem 42 - Algebra 2 and Trigonometry

Question

In \(42-45,\) each polynomial represents the area of a rectangle. Write two binomials that could represent the length and width of the rectangle. $$ 4 x^{2}-7 x-2 $$

Step-by-Step Solution

Answer

The polynomial \(4x^2 - 7x - 2\) factors into \((4x + 1)(x - 2)\).

Step 1: Understanding the Problem

We need to factor the polynomial \(4x^2 - 7x - 2\) into two binomials, since these will represent the length and width of a rectangle. Factoring will involve finding two binomials whose product is the given polynomial.

Step 2: Setup for Factoring by Grouping

The polynomial is a quadratic trinomial of the form \(ax^2 + bx + c\). Here, \(a = 4\), \(b = -7\), and \(c = -2\). We need to find two numbers that multiply to \(a \cdot c = 4(-2) = -8\), and add to \(b = -7\).

Step 3: Finding the Factors

We need two numbers that multiply to \(-8\) and add to \(-7\). Those numbers are \(-8\) and \(+1\).

Step 4: Rewriting the Middle Term

Rewrite \(-7x\) as \(-8x + x\). Therefore, the expression becomes \(4x^2 - 8x + x - 2\).

Step 5: Factoring by Grouping

Group the terms: \((4x^2 - 8x) + (x - 2)\). Factor out the greatest common factor from each group: \(4x(x - 2) + 1(x - 2)\).

Step 6: Extracting the Common Binomial Factor

Notice that \((x - 2)\) is a common factor. Factor it out, obtaining: \((4x + 1)(x - 2)\).

Step 7: Verify the Factorization

To ensure the factorization is correct, expand \((4x + 1)(x - 2)\): \(4x(x) - 4x(2) + 1(x) - 1(2) = 4x^2 - 8x + x - 2 = 4x^2 - 7x - 2\). The factorization is correct.

Key Concepts

Quadratic trinomial

A quadratic trinomial is a polynomial expression consisting of three terms with the highest degree being two. It's represented as \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In our example \(4x^2 - 7x - 2\), the trinomial is composed of the quadratic term \(4x^2\), the linear term \(-7x\), and the constant term \(-2\).

The role of each component in the trinomial is crucial:

• The coefficient \(a\) affects the parabola's direction and width when graphed. If \(a\) is positive, the parabola opens upwards; if negative, it opens downwards.
• The coefficient \(b\) influences the position of the vertex along the x-axis.
• The constant \(c\) determines where the parabola intersects the y-axis.

Quadratic trinomials are essential as they can often represent practical problems such as areas or projectile paths. In our exercise, this trinomial represents the area of a rectangle, which helps us break down the problem into factorable parts.

Factoring by grouping

Factoring by grouping is a technique used to factor polynomials, making it easier to break down and solve complex expressions. We employed this technique in our polynomial \(4x^2 - 7x - 2\).

The technique involves:

• Rewriting the polynomial in four terms instead of three by finding two numbers that both multiply and add together to achieve specific results.
• Grouping these four terms into pairs, making it feasible to factor each group separately.
• Extracting a common factor from each pair, simplifying the expression further.

For instance, 1. We needed numbers multiplying to \(-8\) (\(a \cdot c\)) and adding to \(-7\) (\(b\)). These numbers are \(-8\) and \(+1\). 2. We rewrote \(-7x\) as \(-8x + x\), turning the trinomial into a polynomial of four terms: \(4x^2 - 8x + x - 2\).3. Grouped the expression: \((4x^2 - 8x) + (x - 2)\).4. Factored each group separately: \(4x(x - 2) + 1(x - 2)\).Finally, by recognizing the repeated binomial \((x - 2)\), we could create the fully factored expression \((4x + 1)(x - 2)\). This technique simplifies the polynomial into the product of two binomials, aiding in solving.

Binomials

Binomials are algebraic expressions containing two terms, like \(a + b\). They are the basic building blocks for constructing more complex expressions like polynomials. In this exercise, we factored the quadratic trinomial into two binomials.

These binomials, \((4x + 1)\) and \((x - 2)\), represent the dimensions of a rectangle, or the factors of the area represented by the trinomial. The beauty of binomials is their simplicity, which allows us to reverse-engineer the method used to factor them. Consider these points:

• When expanding binomials, you use the distributive property (FOIL: First, Outer, Inner, Last).
• Understanding binomials forms a cornerstone of algebra, offering insight into solving equations and graphing functions.
• The concept of pairs, which is integral to recognizing factors and products, originates from understanding binomials.

By learning to focus on the structure and relationship within binomials, solving larger polynomial expressions becomes more approachable and intuitive.

Rectangular area representation

In mathematics, representing an area geometrically through algebra is a powerful tool. The given polynomial in our task, \(4x^2 - 7x - 2\), signifies the area of a rectangle, prompting us to factor it to find its dimensions.

Here's why this is useful:

• Polynomials like \(ax^2 + bx + c\) can physically represent shapes, translating abstract algebra into visual geometry.
• Factoring this polynomial into two binomials, \((4x + 1)(x - 2)\), shows the potential length and width of a rectangle.
• Visualizing these derived lengths helps conceptualize how dimensions relate to functional equations.

Understanding algebraic expressions in terms of dimensions simplifies solving for area-based problems. It gives a tangible context to abstract algebra, reinforcing comprehension through visualization. Because many real-world problems are spatial, acknowledging how algebra corresponds to physical geometry broadens our problem-solving arsenal.