Question

Factor completely, by hand or by calculator. Check your results. Trinomials with a Leading Coefficient of 1. $$b^{2}-b-12$$

Short answer

\((b+3)(b-4)\)

Step-by-step solution

Understand the Trinomial

Identify the trinomial as a quadratic expression of the form \(a^2 + bx +c\), where the leading coefficient (the coefficient in front of the highest power of the variable) is 1.

Find the Factors of the Constant Term

List the pairs of factors that multiply to the constant term (-12) in this case. The factor pairs are \(1, -12\), \(2, -6\), \(3, -4\), \(4, -3\), \(6, -2\), and \(12, -1\).

Determine the Correct Pair of Factors

Find the pair of factors that add up to the coefficient of the middle term (-1). This pair is \((3,-4\) since \((3)+(-4)=-1\)).

Write the Binomials

Use the pair of factors to write the trinomial as the product of two binomials: \((b+3)(b-4)\)

Check the Factoring

Expand the binomials to verify the result: \((b+3)(b-4) = b^2 - 4b + 3b - 12 = b^2 - b - 12\) which matches the original expression.

Key concepts

Quadratic Expressions

Quadratic expressions, such as b^2 - b - 12, form the foundation of various algebraic problems. They are polynomial expressions with a degree of 2, which means the highest power of the variable is 2. Structurally, a quadratic expression can be represented as ax^2 + bx + c. The constants a, b, and c are called coefficients, where a is the leading coefficient, b is the linear coefficient, and c is the constant term. These expressions are paramount because they appear in numerous mathematical contexts, such as parabolic graphs, quadratic equations, and many physical applications involving projectile motion and areas of shapes.

Understanding the nature of quadratic expressions enables students to solve a variety of problems, from finding roots or zeros to graphing the parabolic curve. The focus on factoring them leads to simpler, equivalent forms, such as the product of binomials, that can further assist in solving equations or understanding function properties.

Leading Coefficient

The leading coefficient in a polynomial is the coefficient of the term with the highest power of the variable, and it has significant influence on the polynomial's characteristics. In the quadratic expression b^2 - b - 12, the leading coefficient is 1, which simplifies the factoring process. When the leading coefficient is 1, it's known as a 'monic' quadratic, and factoring methods can be more straightforward since you only need to consider the constant term when looking for factors.

For non-monic quadratics, where the leading coefficient is not 1, additional steps are needed to factor correctly - sometimes involving methods like the grouping method or the use of the quadratic formula. The leading coefficient also affects the direction and width of the parabola when graphed, making it a vital concept when transitioning from algebraic manipulation to graphical representation.

Binomial Products

When factoring quadratic expressions, particularly those with a leading coefficient of 1, our goal is to express them as products of binomials. A binomial is a two-termed algebraic expression, and its product with another binomial follows the FOIL method (First, Outer, Inner, Last) to yield a quadratic expression. For instance, the product of (b + m)(b + n) results in b^2 + (m+n)b + mn, where m and n are the factors of the constant term in the original trinomial.

The factoring process involves finding two numbers that multiply to give the constant term and add to give the middle term's coefficient. It’s like a puzzle requiring the correct combination to fit. Once the correct pair of numbers is found, as shown in the original problem's solution, they are used to construct the binomials that, when multiplied, give back the original quadratic expression. This conversion from trinomial to binomial products is essential in simplifying complex algebraic expressions and solving quadratic equations.

Algebraic Factorization Methods

Algebraic factorization methods turn complex expressions into products of simpler expressions, thus breaking down the problem into more manageable pieces. Besides factoring trinomials with a leading coefficient of 1, other methods include factoring by grouping, factoring out the greatest common factor (GCF), using special products like the difference of squares, and factoring sum or difference of cubes.

Each method has its specific use based on the form of the algebraic expression. When trinomials can't be factored easily, the quadratic formula may come to the rescue. Mastery of these methods allows students to tackle a wide range of algebraic challenges, making understanding how and when to apply them quite important. In essence, factorization is a vital tool, providing a pathway from complex, unwieldy expressions to streamlined, solvable equations.