Question

Factor completely, by hand or by calculator. Check your results. Trinomials with a Leading Coefficient of 1. $$x^{2}-15 x+56$$

Short answer

\(x^2 - 15x + 56 = (x - 7)(x - 8)\)

Step-by-step solution

Determine the factors of the constant term

Identify the factors of 56 that add up to the middle coefficient, which is -15. The pair of numbers that multiply to 56 and add up to -15 are -7 and -8.

Write the factorization

Express the trinomial as a product of two binomials using the factors found in Step 1: \((x - 7)(x - 8)\).

Check the result through FOIL

Multiply the two binomials using the FOIL method to verify the product matches the original expression: First, Outer, Inner, Last.\(x \cdot x = x^2\), \(x \cdot (-8) = -8x\), \((-7) \cdot x = -7x\), and \((-7) \cdot (-8) = 56\).Combine like terms: \(x^2 - 8x - 7x + 56\) simplifies to \(x^2 - 15x + 56\), which matches the original trinomial.

Key concepts

Leading Coefficient

The leading coefficient in a polynomial is the coefficient of the term with the highest degree. For instance, in the trinomial \(x^2-15x+56\), the leading coefficient is 1, as the term with the highest degree, \(x^2\), has a coefficient of 1. When factoring trinomials with a leading coefficient of 1, the task is simplified because we only need to find two numbers that multiply to the constant term (in this case, 56) and simultaneously add to the linear term's coefficient (-15).

Understanding the role of the leading coefficient is crucial because it can change the factorization technique applied. For trinomials with a leading coefficient other than 1, methods like trial and error or the ac method may become necessary.

FOIL Method

The FOIL method is a shortcut to multiply two binomials and stands for First, Outer, Inner, and Last. This mnemonic reminds us of the order in which we should distribute the terms. For the product of \(x-7\) and \(x-8\), FOIL instructs us to multiply in the following steps:

• First: Multiply the first terms in each binomial (\(x\) by \(x\)).
• Outer: Multiply the outer terms (\(x\) by -8).
• Inner: Multiply the inner terms (-7 by \(x\)).
• Last: Multiply the last terms (-7 by -8).

Afterward, we combine like terms to check if we get back the original trinomial. The FOIL method is at the heart of understanding binomial products and is essential for verifying factorizations.

Binomial Products

Binomial products result from multiplying two binomials together, often involving the FOIL method. For example, when we multiply the binomials \(x-7\) and \(x-8\), we obtain the binomial product \(x^2-15x+56\). This product is the trinomial we started with, confirming our factorization's accuracy.

Binomial products are foundational in algebra and appear frequently across various mathematical operations. They are fundamental blocks for constructing and deconstructing polynomial expressions. Mastering binomial multiplication fosters a deep understanding of algebraic relationships and simplifies complex polynomial operations.

Factorization Techniques

Factorization techniques refer to the methods used to break down a polynomial into a product of simpler polynomials or numbers, known as factors. As seen in our example, for trinomials with a leading coefficient of 1, we look for two binomials whose product returns the original polynomial. The process usually involves identifying a pair of numbers that both add to the linear coefficient and multiply to the constant term.

Other factorization techniques include grouping, which we use when dealing with polynomials of higher degrees or when the leading coefficient is not 1, and the use of the quadratic formula in certain trinomials. As students progress in algebra, they will encounter special products, such as difference of squares and perfect square trinomials, which each require their own specialized approach to factorization.