Problem 5 - Elementary and Intermediate Algebra

Question

If \(10 x^{2}-27 x+5\) is to be factored as the product of two binomials, what are the possible first terms of the binomial factors?

Step-by-Step Solution

Answer

The possible first terms are 1, 10, 2, and 5.

Step 1: Understand the General Form

The expression to factor is a quadratic, and we aim to factor it into two binomials of the form \((ax + b)(cx + d)\) that equal \(10x^2 - 27x + 5\). The first terms of each binomial will multiply to give the leading coefficient, which is 10.

Step 2: Determine Possible First Terms

To factor \(10x^2\), the first terms in the binomials must be pairs of factors of 10: \((1x)(10x)\), \((2x)(5x)\), or any interchange of these pairs. This means the possible first terms for each binomial could be 1, 10, 2, or 5.

Key Concepts

Binomial Factors

In the world of algebra, a *binomial* is an algebraic expression that contains two terms. When we talk about *binomial factors*, we're referring to expressions such as \((ax + b)(cx + d)\) where each factor is a binomial. The process of breaking down a quadratic expression into binomial factors involves finding two expressions that can be multiplied together to yield the original quadratic.For example, in the given quadratic \(10x^2 - 27x + 5\), our goal is to express this as the product of two binomials. These binomials take the form \((px + m)(qx + n)\), where \(p\) and \(q\) are the first terms. Understanding how these binomials are structured is crucial for effective factoring. The terms of the binomials will multiply in a specific way to recreate each part of the quadratic expression.

Leading Coefficient

The *leading coefficient* is a key player in factoring quadratics. It refers to the coefficient of the highest degree term in the polynomial, providing essential clues as to how the polynomial can be factored.In the expression \(10x^2 - 27x + 5\), 10 is the leading coefficient. When factoring this quadratic into binomials, the first terms of these binomials need to multiply together to equal 10. This makes the leading coefficient a guide for the possible first terms of the binomial factors.Understanding the leading coefficient is crucial. It influences the choices we make in our first terms. It helps narrow down the possibilities from an infinite number to a finite, manageable set of values. Adjusting these values correctly will ensure that the factoring process replicates the original expression accurately.

Pairs of Factors

When solving quadratics, identifying the *pairs of factors* of the leading coefficient is a vital step. We need to determine which pairs of numbers can multiply together to recreate the leading coefficient.For example, with a leading coefficient of 10, the valid pairs for \(10x^2\) are \((1x)(10x)\) and \((2x)(5x)\). These pairs stem from the factors of 10, such as 1 and 10, or 2 and 5. Analyzing these pairs helps us determine potential first terms for the binomial factors.

• 1 and 10 form one possible pair.
• 2 and 5 form another viable pair.

Once you have identified these pairs, factorizing becomes a process of elimination, matching these pairs with their corresponding terms and checking through multiplication that you regain the original quadratic expression.Understanding pairs of factors enables a systematic approach to finding the accurate binomial factors efficiently.