Problem 105 - Algebra 1: Concepts and Skills

Question

Factor the trinomial. (Lessons 10.5,10.6) $$ 2 x^{2}-x-3 $$

Step-by-Step Solution

Answer

The factorization of the given trinomial \(2x^2-x-3\) is \((2x + 1)(x - 3)\).

Step 1: Identify the Coefficients

We have a trinomial of the form \(ax^2+bx+c\). Identify the coefficients \(a = 2, b = -1, c = -3\). Next, find two numbers m and n that add up to \(b\), and when multiplied give \(ac=(-6)\)

Step 2: Finding two Numbers

Now find the two numbers \(m, n\) to satisfy the relationship: \(m+n=b, mn=ac\). The two numbers that meet these requirements are \(-3\) and \(2\). As their sum equals to \(b=-1\) and their product is \(ac=-6\).

Step 3: Factoring the Trinomial

Using these values and having the numbers \(m, n\), we rewrite the trinomial as a product of binomials. Therefore, the trinomial \(2x^2-x-3\) factors as \((2x + 1)(x - 3)\).

Key Concepts

Quadratic Equations

Quadratic equations are foundational in algebra. They are polynomial equations of the second degree and have the general form \(ax^2 + bx + c = 0\). Here, \(x\) represents an unknown variable, and \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). This is because if \(a\) were zero, the equation would no longer be quadratic but linear.
Understanding quadratic equations is crucial because they help describe various physical phenomena and solve algebraic problems. They typically have two solutions or roots, which can be real or complex numbers.
Solving quadratic equations can involve multiple methods:

• Factoring the quadratic expression.
• Completing the square.
• Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\).

It's important for students to become familiar with these methods as they develop their mathematical skills.

Coefficient Identification

Identifying coefficients is the first critical step in solving or factoring quadratic equations. Each term in a quadratic equation is made up of constituents, and these include:

• \(a\): the coefficient of \(x^2\).
• \(b\): the coefficient of \(x\).
• \(c\): the constant term.

In our original exercise, the trinomial \(2x^2 - x - 3\) has:

\(a = 2\)

\(b = -1\)

\(c = -3\)

Coefficient identification enables you to understand the structure of the quadratic equation and dictates the approach you will use to factor or solve it. For example, these coefficients are vital when using methods like factoring by grouping or applying the quadratic formula.

Polynomial Factoring

Polynomial factoring is a method used to simplify or solve polynomial equations by expressing them as the product of simpler polynomials. In the case of quadratic trinomials, this means rewriting them as the product of two binomials.
To factor a polynomial, especially a quadratic trinomial, follow these steps:

• Identify the coefficients \(a\), \(b\), and \(c\).
• Calculate the product \(ac\) and find two numbers \(m\) and \(n\) that, multiplied together, equal \(ac\) and add up to \(b\).
• Reformulate the middle term (\(bx\)) using \(m\) and \(n\).
• Factor by grouping, rewriting the expression as a product of binomials.

Although this might sound complex, with practice, identifying these numbers and reformulating the expression becomes intuitive.

Trinomial Products

Trinomial products are the result of multiplying two binomials. To gain a deeper understanding, recognizing how these products relate to factors is crucial.
When you expand a product of two binomials, \((px + q)(rx + s)\), it returns a trinomial \(prx^2 + (ps + qr)x + qs\). To factor a trinomial back into binomials, reverse this process:

• Check for a common factor in all terms.
• If none, identify two numbers that fit the sum-product requirement discussed in polynomial factoring.
• Rewrite the middle term using these numbers, and regroup.
• Factor by grouping to revert it back into the original binomial form.

In factoring \(2x^2 - x - 3\), this process helped us arrive at the factors \((2x + 1)(x - 3)\). By understanding this transformation, students can reverse-engineer trinomials back into products, simplifying problem-solving. The mastery of trinomial products enables students to handle varying polynomial problems efficiently.