Problem 81 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x+c\). \(p^{2}+5 p-6\)

Step-by-Step Solution

Answer

\( (p + 6)(p - 1) \)

Step 1: Identify coefficients

Identify the coefficients from the trinomial expression. Here, we have the expression \( p^2 + 5p - 6 \). The coefficient of \( p^2 \) is 1, the coefficient of \( p \) is 5, and the constant term is -6.

Step 2: Find two numbers that multiply to c and add to b

Look for two integers that multiply to \( -6 \) (the constant term) and add to 5 (the coefficient of \( p \)). These numbers are 6 and -1 because \( 6 \times (-1) = -6 \) and \( 6 + (-1) = 5 \).

Step 3: Write the factored form

Using the numbers identified in the previous step, write the trinomial as a product of two binomials. In this case, \( p^2 + 5p - 6 = (p + 6)(p - 1) \).

Key Concepts

factoring techniques

Factoring techniques refer to various methods used to break down algebraic expressions into simpler components called factors. In the context of trinomials, we often break them down into binomial products. One effective method is factoring by inspection, especially for trinomials in the form of \(x^2 + bx + c\). Here are the key steps involved:

• Identify the coefficients and constant term.
• Find two numbers that multiply to the constant term and add to the middle coefficient.
• Use these numbers to rewrite the trinomial as a product of two binomials.

By following these steps, you can simplify complex polynomial expressions and solve quadratic equations more efficiently.

trinomial expressions

A trinomial expression is a type of polynomial that consists of three terms. The general form is \(ax^2 + bx + c\), where \(a, b\), and \(c\) are constants, and \(x\) is the variable. In simpler terms, a trinomial is made up of:

• A term with the variable squared (quadratic term).
• A term with the variable to the first power (linear term).
• A constant term (no variable).

For example, in the given exercise, \(p^2 + 5p - 6\) is a trinomial where \(p^2\) is the quadratic term, \(5p\) is the linear term, and \(-6\) is the constant. Understanding this structure is crucial when learning to factor and simplify these expressions.

mathematical coefficients

Coefficients are numbers that multiply the variables in algebraic expressions. In a trinomial like \(p^2 + 5p - 6\), we have:

• The coefficient of the quadratic term \(p^2\) is 1.
• The coefficient of the linear term \(5p\) is 5.
• The constant term is \(-6\).

Recognizing these coefficients helps in identifying the parts of the expression correctly and is essential when applying factoring techniques. You use them to find suitable pairs of factors that will help you break down the trinomial into binomial products.

binomial products

Binomial products are the result of multiplying two binomials. For example, in the given exercise, we factored \(p^2 + 5p - 6\) into \((p + 6)(p - 1)\). This means:

• When you multiply \(p + 6\) and \(p - 1\),
• You get back the original trinomial \(p^2 + 5p - 6\).

The reverse process, starting from a trinomial and splitting it, involves finding two numbers that sum to the middle coefficient and multiply to the constant term. In this case, finding 6 and \(-1\) made it possible to write the trinomial as a product of binomials, simplifying problem-solving in algebra.