Problem 34 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(x^{2}+3 x-10\)

Step-by-Step Solution

Answer

The expression is factored as \((x + 5)(x - 2)\).

Step 1: Identify the Coefficients

For the quadratic expression \(x^2 + 3x - 10\), identify the coefficients: \(a = 1\), \(b = 3\), and \(c = -10\). These correspond to the general form \(ax^2 + bx + c\).

Step 2: Find Two Numbers that Multiply to c and Add to b

We need two numbers that multiply to \(-10\) (the constant \(c\)) and add up to \(3\) (the linear coefficient \(b\)). The numbers are \(5\) and \(-2\) because \(5 \times (-2) = -10\) and \(5 + (-2) = 3\).

Step 3: Rewrite the Middle Term

Rewrite the middle term \(3x\) as the sum of two terms using the numbers found in Step 2: \(x^2 + 5x - 2x - 10\).

Step 4: Group the Terms

Group the terms into two pairs: \((x^2 + 5x)\) and \((-2x - 10)\).

Step 5: Factor by Grouping

Factor out the greatest common factor from each pair. From \(x^2 + 5x\), factor \(x\): \(x(x + 5)\). From \(-2x - 10\), factor \(-2\): \(-2(x + 5)\).

Step 6: Extract the Common Binomial Factor

Notice that \((x + 5)\) is common in both terms. Factor \((x + 5)\) out: \((x + 5)(x - 2)\).

Step 7: Verify the Factorization

Expand the factored expression to verify correctness: \((x + 5)(x - 2) = x^2 - 2x + 5x - 10 = x^2 + 3x - 10\). The original expression is regained, confirming the factorization is correct.

Key Concepts

Factoring Trinomials

Factoring trinomials is an essential skill in algebra that involves expressing a trinomial as the product of two binomials. Trinomials are polynomials with three terms, commonly structured as \(ax^2 + bx + c\). The goal of factoring is to simplify or rewrite expressions to solve equations or to make them easier to handle in calculations.

• Step 1: Identify the coefficients \(a\), \(b\), and \(c\) from the trinomial. These are the leading coefficient, the middle coefficient, and the constant term, respectively.
• Step 2: Look for two numbers that multiply to the constant term \(c\) and add to the middle coefficient \(b\). This is the key observation to factor correctly.
• Step 3: Rewrite the trinomial using these two numbers to substitute the middle term, which helps to reveal a pattern for grouping.

Factoring involves practice and observation. With experience, you'll get quicker at spotting potential pairs that meet the criteria for multiplication and addition.

Algebraic Expressions

An algebraic expression is a mathematical phrase that involves numbers, variables, and operation symbols like addition and multiplication. In the context of factoring, focusing on algebraic expressions refers to the simplification or transformation of a polynomial into a product of simpler expressions.

• They can represent real-world scenarios and are used extensively to model equations and inequalities.
• The expression \(x^2 + 3x - 10\) is an example where we use variables and constants together in terms of operations.
• Breaking down these into partial expressions helps in realizing factorization by grouping terms that share a common factor.

Simplifying algebraic expressions through factorization allows us to solve complex equations more efficiently. This process is exceptionally helpful when solving quadratic equations.

Polynomial Equation Solving

Solving polynomial equations often involves transforming the equation into a factored form. In this way, the roots or solutions can be readily identified. This process directly uses the concept of factoring trinomials.

• When a polynomial is expressed as a product of simpler polynomials, setting each factor equal to zero allows us to solve for the variable.
• In our example, once we represent \(x^2 + 3x - 10\) as \((x + 5)(x - 2)\), setting each binomial to zero gives logical roots: \(x + 5 = 0\) or \(x - 2 = 0\).
• This means \(x = -5\) or \(x = 2\) are our solutions. Identifying these solutions are crucial in understanding the behavior of quadratic relationships in functions.

By mastering polynomial equation solving, students can better tackle a wide range of mathematical problems, especially those involving average-rate changes, projectile motion, and parabolic graphs.