Problem 125 - Algebra and Trigonometry Real Mathematics, Real People

Question

Factor the trinomial. $$2 x^{2}+11 x-40$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(2x^2+11x-40\) is \((2x-5)(x+8)\).

Step 1: Identify the Coefficients

The given trinomial is in the form \(ax^2 + bx + c\). Here, \(a = 2\), \(b = 11\), and \(c = -40\).

Step 2: Find two numbers

Find two numbers that add up to \(b = 11\) and multiply up to \(ac = 2*(-40) = -80\). These numbers are 16 and -5, because \(16 + (-5) = 11\) and \(16 * (-5) = -80\).

Step 3: Rewrite the Trinomial

Rewrite the middle term of the trinomial as the sum of the terms 16x and -5x. This gives us the expression: \(2x^2 + 16x - 5x - 40\).

Step 4: Factor by Grouping

Regroup the terms to factor by grouping: \(2x(x+8) -5(x+8)\). This results in the factored form: \((2x-5)(x+8)\).

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation represented by the general form \(ax^2 + bx + c = 0\). These equations are characterized by the highest degree of the variable being 2, hence the name "quadratic," which comes from the Latin word for "square." Understanding quadratic equations is foundational because they frequently appear in algebraic contexts and applications.
Here are some vital aspects to know:

• The solutions or roots of a quadratic equation can be found using various methods, including factoring, completing the square, or using the quadratic formula \(-\frac{b \pm \sqrt{b^2 - 4ac}}{2a}\).
• The parabola, which is the graph of a quadratic equation, can open upwards or downwards depending on the sign of the coefficient \(a\).
• The quadratic equation is fully described by its coefficients \(a\), \(b\), and \(c\), with \(a eq 0\).

Using these concepts, students can interpret and solve quadratic equations efficiently.

Coefficient Identification

In any quadratic equation, identifying coefficients is a crucial first step in the problem-solving process. The equation typically takes the form \(ax^2 + bx + c = 0\). Each letter represents a numerical coefficient that impacts the behavior and solutions of the equation.
Coefficients are identified as follows:

• \(a\) is the coefficient of \(x^2\) and determines the parabola's width and direction (upward or downward).
• \(b\) is the coefficient of \(x\) and affects the symmetry and location of the parabola.
• \(c\) is the constant term, influencing the vertical shift of the parabola.

Knowing these coefficients allows you to identify and appropriately manipulate them during the factoring process or when applying other algebraic methods to solve or simplify the equation.

Factor by Grouping

Factor by grouping is a technique used in polynomial factorization, especially when dealing with quadratic trinomials that do not factor easily by basic inspection. The ultimate goal is to rearrange and group terms to reveal common factors, thereby simplifying the expression.
The process typically involves:

• Rewriting the middle term \(bx\) as the sum of two terms whose coefficients multiply to produce \(ac\) (the product of \(a\) and \(c\)), and add to \(b\).
• Splitting the expression into two groups based on the rewritten terms.
• Factoring out the greatest common factor from each group.
• Identifying and externalizing any common binomial factor, yielding the fully factored form of the polynomial.

Through this systematic approach, factor by grouping allows for the efficient breakdown and resolution of complex polynomial equations.

Polynomial Factorization

Polynomial factorization is a fundamental algebraic technique used to express a polynomial as the product of simpler polynomials. When dealing with trinomials like \(2x^2 + 11x - 40\), the objective is to break them down into two binomial factors.
Key points in factorization include:

• Understanding that the process involves reversing the distributive property, starting with identifying suitable pairs of terms that reconstruct the original polynomial when multiplied.
• Employing strategies like the identification of coefficients and numbers that satisfy multiplication and addition conditions from the polynomial's terms.
• Applying specific methods, such as factor by grouping, when simple factorization isn't immediately obvious.

By mastering polynomial factorization, students gain powerful tools for solving equations and simplifying algebraic expressions, enhancing both problem-solving skills and algebraic understanding.