Problem 25 - College Algebra

Question

In Exercises \(17-30,\) factor each trinomial, or state that the trinomial is prime. $$3 x^{2}-25 x-28$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(3x^2 - 25x - 28\) is \((3x + 4)(x - 7)\).

Step 1: Identify a, b, and c

In a quadratic trinomial \(ax^2 + bx + c\), identify the coefficients of the terms. Here, \(a = 3\), \(b = -25\), and \(c = -28\).

Step 2: Find two numbers

Find two numbers, let's call them p and q, such that :\n1) They multiply together to give the product \(ac\), which is \(3*-28 = -84\), and\n2) They add up to form \(b\), which is -25.\nThe numbers that satisfy these conditions are -21 and 4 as \(-21*4=-84\) and \(-21+4 = -25\).

Step 3: Rewrite the trinomial

Rewrite the trinomial, splitting the middle term using the p and q values. So, \(3x^2 - 25x - 28\) is equivalent to \(3x^2 -21x + 4x - 28\).

Step 4: Factorise by grouping

Group the first two terms together and the last two terms together, and factor each group. This yields: \(3x(x - 7) + 4(x - 7)\).

Step 5: Write the final factorised form

Notice that \((x - 7)\) is a common factor now. The final factored form is \((3x + 4)(x - 7)\).

Key Concepts

Quadratic Trinomials

Quadratic trinomials are algebraic expressions of the form ax^2 + bx + c, where a, b, and c are constants, and x represents a variable. These expressions are standard in algebra, and learning to factor them is an essential skill for solving quadratic equations. Factoring such an expression means expressing it as a product of two binomials. For instance, the quadratic trinomial 3x^2 - 25x - 28 can be factored down to (3x + 4)(x - 7), making its solution much more manageable.

In many cases, quadratic trinomials can be factored easily when a (the coefficient of x^2) is 1. However, when a is greater than 1, the process can become more intricate, but with the understanding of certain strategies such as 'factor by grouping', it can be approached systematically.

A standard method includes searching for two numbers that both add up to b and multiply to a*c. If such numbers can be found, the trinomial is factorable; if not, the trinomial is considered to be 'prime' and cannot be factored over the set of integers.

Factor by Grouping

Factor by grouping is a technique used in factoring polynomials when direct factoring is not apparent. The process involves rearranging the terms of the polynomial and grouping them in such a way that the common factors can be factored out. In the context of quadratic trinomials, after identifying two numbers p and q, which respectively multiply to ac and add up to b, these numbers are used to split the middle term bx.

By dividing the trinomial into two groups, we can then factor out the common factors from each group. In our example, 3x^2 -21x and + 4x - 28 were grouped separately, which both contain a common binomial (x - 7). This technique simplifies the trinomial by allowing us to factor out this binomial and complete the factoring process.

Coefficient Identification

Coefficient identification is the preliminary step in factoring, where you recognize and label the coefficients in a quadratic trinomial ax^2 + bx + c. These coefficients play crucial roles in determining the factorability of the trinomial and in applying the appropriate factoring technique.

Understanding Coefficients

In the example 3x^2 -25x -28, the coefficient a is 3, b is -25, and c is -28.

• a influences the shape and orientation of the parabola when the trinomial is graphed.
• b influences the position of the vertex of the parabola along the horizontal axis.
• c gives the y-intercept of the parabola.

Having a strong grasp of how these coefficients affect the polynomial is crucial for factoring, as their signs and magnitudes affect the numbers you will choose during the 'factor by grouping' steps.

Algebraic Factorization

Algebraic factorization is the process of breaking down complex algebraic expressions into products of simpler factors. This is advantageous for simplifying expressions and solving equations. In the context of quadratic trinomials, algebraic factorization allows us to convert an expression like 3x^2 - 25x - 28 into a product of two binomials (3x + 4) and (x - 7).

Factorization enables us to locate the roots of the quadratic equation (i.e., the values of x for which the trinomial equals zero) easily. These roots correspond to the solutions of the equation ax^2 + bx + c = 0, and finding them is a common objective in algebra. Once a trinomial is factored, the Zero Product Property can be applied, stating that if a product of factors equals zero, then at least one of the factors must be zero. This is the guiding principle that allows us to solve for x once the trinomial is fully factored.