Problem 79 - College Algebra and Calculus: An Applied Approach

Question

Describe two different ways to factor \(2 x^{2}-7 x-15\).

Step-by-Step Solution

Answer

Two ways to factor \(2x^2-7x-15\) are \((2x+3)(x-5)\) or \((2x-15)(x-1)\).

Step 1: Trinomial Structure

Observe that the given trinomial is of the form \(ax^2+bx+c\), with \(a=2\), \(b=-7\), and \(c=-15\). Identify the terms and their respective coefficients in the quadratic.

Step 2: Identify Pairs

Identify pairs of numbers that multiply to give a product of \(ac\) (which is \(2*(-15)=-30\)) and sum to \(b\) (which is -7). After some guesses and checks, these pairs are found to be: (-10 and 3) or (-15 and 2).

Step 3: Decompose and Factor

Decompose the middle term into the identified pairs: \(2x^2-10x+3x-15\) or \(2x^2-15x+2x-15\), and factor by grouping.

Step 4: Quadratic Expressions

For the pairs\((-10, 3)\), it factors to \(2x(x-5)+3(x-5)=(2x+3)(x-5)\), and for the pairs (-15, 2), it factors to \((2x^2-15x)+(2x-15)=x(2x-15)-1(2x-15)=(2x-15)(x-1)\).

Key Concepts

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree two. They have the standard form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The graph of a quadratic expression is a parabola, which can either open upwards or downwards, depending on the sign of \(a\). In our exercise, we are given the quadratic expression: \(2x^2 - 7x - 15\). Here, \(a = 2\), \(b = -7\), and \(c = -15\).
Understanding the coefficients is crucial as they dictate the shape and position of the parabola, as well as the solutions of the quadratic equation.

• The coefficient \(a\) influences the width and the direction of the parabola.
• The coefficient \(b\) affects the axis of symmetry and the turn direction.
• Finally, \(c\) represents the y-intercept when \(x = 0\).

Factor by Grouping

Factor by grouping is a method used to factorize quadratic expressions, particularly when dealing with trinomials. This technique involves rearranging the terms in a polynomial and grouping them in pairs that can be easily factored separately.
For our quadratic \(2x^2 - 7x - 15\), we start by finding pairs of numbers that multiply to the product of \(a\) and \(c\) (in this case, \(-30\)) and add to \(b\) (which is \(-7\)). After trying different combinations, we find that the pairs \((-10, 3)\) and \((-15, 2)\) satisfy these conditions.
We then break down the middle term, \(-7x\), into these pairs:

• \(2x^2-10x+3x-15\)
• \(2x^2-15x+2x-15\)

Both equations are then grouped to form two pairs:

• \((2x^2-10x) + (3x-15)\)
• \((2x^2-15x) + (2x-15)\)

Factoring out the greatest common factor from each group and combining them as products will give us the factors of the original quadratic expression.

Trinomial Structure

Trinomial structure is a specific format within quadratic expressions and is vital for understanding how to factor them efficiently. A trinomial expression has three terms, and often, it appears in the standard form \(ax^2 + bx + c\). The factorization process involves transforming this structure into a product of two binomials.
The trinomial \(2x^2 - 7x - 15\) can be transformed by identifying a pair of numbers that fit the criteria of both multiplication and addition.

• Multiply \(a\) and \(c\) to find pairs that add up to \(b\).
• Use those pairs to break down and rearrange the terms, making perfect binomial factors easier to spot.

In practice, this might look like using the pairs \((-10, 3)\) or \((-15, 2)\) to split the middle term. Consequently, this sets up the trinomial for grouping and factoring into simpler binomial expressions. Each of these steps ensures you're working towards simplifying the trinomial into a product of two binomials, making solutions to the equation cleaner and solving it more efficiently.