Problem 31 - Algebra 1: Concepts and Skills

Question

Factor the trinomial. $$ 2 z^{2}+19 z-10 $$

Step-by-Step Solution

Answer

The factored form of the trinomial \(2z^2 + 19z - 10\) is \((2z - 1)(z + 10)\).

Step 1: Identification

Identify a, b, and c in the equation from the standard quadratic equation \(ax^2 + bx + c\). In this equation, \(a = 2\), \(b = 19\), and \(c = -10\).

Step 2: Find two numbers

We need to find two numbers that multiply to \(-20\) (the product of \(a\) and \(c\)) and add to 19 (the coefficient of \(z\)). After examination, we discover that these numbers are 20 and -1 because \(20 × -1 = -20\) (which is \(ac\)) and \(20 + -1 = 19\) (which matches with \(b\)).

Step 3: Rewrite the middle term

We rewrite the middle term of the equation \(2z^2 + 19z - 10\) as (20z - z). This leads to \(2z^2 + 20z - z - 10\).

Step 4: Factor by grouping

Now we can factor by grouping. Group the first two terms together and the last two terms together: \(2z(z + 10) - 1(z + 10)\). From this step, factorize \(z+10\) from both terms to get \((2z - 1)(z + 10)\).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra, forming the backbone of various mathematical applications. A quadratic equation is any equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). These equations tell us about parabolas when graphed, which can open upwards or downwards depending on the sign of \(a\).

Understanding quadratic equations involves mastering how to solve them either by factoring, using the quadratic formula, or completing the square. Let's talk specifically about factoring. This means expressing the quadratic in a product form like \((px + q)(rx + s) = 0\). Finding the roots (or solutions) of the equation involves setting each factor equal to zero, because for a product to be zero, one of the factors must be zero. This concept helps solve for unknown variables.

When you're solving a quadratic equation, the first step is to make sure it's in standard form \(ax^2 + bx + c = 0\). Once in this form, you can decide which method to use for solving it, tailoring your approach based on what the equation looks like. Factoring is always a good first try if the constants are integers.

Factoring by Grouping

Factoring by grouping is a powerful method to factor trinomials when simple extraction isn't clear straight away. It involves splitting the middle term in such a way that you can form two binomials and then factor them out separately.

Let's break it down using our example trinomial \(2z^2 + 19z - 10\). We first identify that we need two numbers that multiply to \(a \times c\) and sum to \(b\), which here are 20 and -1. We rewrite the middle term, reshaping the expression into something like \(2z^2 + 20z - z - 10\).

Next, we group terms: \((2z^2 + 20z) + (-z - 10)\). In each group, look for a common factor. From the first pair, \(2z\) can be factored out; from the second, factor out \(-1\), giving us \(2z(z + 10) - 1(z + 10)\).

The final step of factoring by grouping is making use of the common factor, which in this case is \(z + 10\). Observing this factor common to both terms, we factor it out to get \((2z - 1)(z + 10)\). This method can seem a bit tricky at first, but with consistency and practice, it becomes a straightforward process.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operations, and they are the building blocks for equations. They are different from equations because they don’t have an "equals" sign. Instead, think of them as mathematical phrases that can represent a wide range of values.

Operations within algebraic expressions include addition, subtraction, multiplication, and division, alongside exponents which can affect the variables involved. Simplifying expressions is a common task, ensuring they are in the simplest or most useful form.

When factoring providing expressions like the quadratic \(2z^2 + 19z - 10\), you're breaking down the algebraic expression into simpler components (factors), which makes finding solutions to related equations much easier. This breaking down process not only aids in solving quadratic equations but is also foundational in higher mathematics involving polynomials and complex algebraic landscapes.

Moreover, understanding algebraic expressions and how to manipulate them beyond just factoring is key in fields such as engineering and physics, where these mathematical phrases describe real-world systems and phenomena. The power of algebraic expressions lies in their generality and ability to adapt to model numerous situations.