Factor X^2+5x-6

[Diante Scharsch]

Noticethat in Example 4b, the sign of each term is changed when the expression is written without parentheses. This is the same result that we would have obtained if we used the procedures that we introduced in Section 2.5 to simplify expressions.

Thus, if there is a monomial factor common to all terms in a polynomial, we can write the polynomial as the product of the common factor and another polynomial. For instance, since each term in x2 + 3x contains x as a factor, we can write the expression as the product x(x + 3). Rewriting a polynomial in this way is called factoring, and the number x is said to be factored "from" or "out of' the polynomial x2 + 3x.

To factor a monomial from a polynomial:Write a set of parentheses preceded by the monomial common to each term in the polynomial.Divide the monomial factor into each term in the polynomial and write the quotient in the parentheses.Generally, we can find the common monomial factor by inspection.

In this book, we will restrict the common factors to monomials consisting of numerical coefficients that are integers and to integral powers of the variables. The choice of sign for the monomial factor is a matter of convenience. Thus,

We can use the distributive law to multiply two binomials. Although there is little need to multiply binomials in arithmetic as shown in the example below, the distributive law also applies to expressions containing variables.

With practice, you will be able to mentally add the second and third products. Theabove process is sometimes called the FOIL method. F, O, I, and L stand for: 1.The product of the First terms.2.The product of the Outer terms.3.The product of the Inner terms.4.The product of the Last terms.

In Section 4.3, we saw how to find the product of two binomials. Now we will reverse this process. That is, given the product of two binomials, we will find the binomial factors. The process involved is another example of factoring. As before,we will only consider factors in which the terms have integral numerical coefficients. Such factors do not always exist, but we will study the cases where they do.

Note that when all terms of a trinomial are positive, we need only consider pairs of positive factors because we are looking for a pair of factors whose product and sum are positive. That is, the factored term of

When the first and third terms of a trinomial are positive but the middle term is negative, we need only consider pairs of negative factors because we are looking for a pair of factors whose product is positive but whose sum is negative. That is,the factored form of

Solution

We look for two integers whose product is 12 and whose sum is 5. From the table in Example 1 on page 149, we see that there is no pair of factors whose product is 12 and whose sum is 5. In this case, the trinomial is not factorable.

Skill at factoring is usually the result of extensive practice. If possible, do the factoring process mentally, writing your answer directly. You can check the results of a factorization by multiplying the binomial factors and verifying that the product is equal to the given trinomial.

In Section 4.4 we factored trinomials of the form x2 + Bx + C where the second-degree term had a coefficient of 1. Now we want to extend our factoring techniquesto trinomials of the form Ax2 + Bx + C, where the second-degree term has acoefficient other than 1 or -1.

The trinomial 4x2 - 5x + 3 is not factorable, since the above table shows thatthere is no pair of factors whose product is 12 and whose sum is -5. The test tosee if the trinomial is factorable can usually be done mentally.

Once we have determined that a trinomial of the form Ax2 + Bx + C is fac-torable, we proceed to find a pair of factors whose product is A, a pair of factorswhose product is C, and an arrangement that yields the proper middle term. Weillustrate by examples.

1. We consider all pairs of factors whose product is 4. Since 4 is positive, only positive integers need to be considered. The possibilities are 4, 1 and 2, 2.

2. We consider all pairs of factors whose product is 3. Since the middle term is positive, consider positive pairs of factors only. The possibilities are 3, 1. We write all possible arrangements of the factors as shown.

With practice, you will be able to mentally check the combinations and will notneed to write out all the possibilities. Paying attention to the signs in the trinomialis particularly helpful for mentally eliminating possible combinations.

Often we must solve equations in which the variable occurs within parentheses. Wecan solve these equations in the usual manner after we have simplified them byapplying the distributive law to remove the parentheses.

Sometimes, the mathematical models (equations) for word problems involveparentheses. We can use the approach outlined on page 115 to obtain the equation.Then, we proceed to solve the equation by first writing equivalently the equationwithout parentheses.

In this section, we will examine several applications of word problems that lead toequations that involve parentheses. Once again, we will follow the six steps out-lined on page 115 when we solve the problems.

The basic idea of problems involving coins (or bills) is that the value of a numberof coins of the same denomination is equal to the product of the value of a singlecoin and the total number of coins.

This calculator is a free online math tool that transforms polynomial into factored form.The calculator can be used to factor polynomials having one or more variables.Calculator shows all the work and provides detailed explanation how tofactor the expression.

Benjamin is designing a new house. The bedroom closet will have one wall that contains a closet system using three different-sized storage units. The number and amount of wall space needed for each of the three types of storage units is shown. What are the dimensions of the largest amount of wall space that will be needed?

Compute: Because the area of the rectangle is the product of the length and the width, factor the expression to find binomials that represent the length and the width of the closet wall.

Factorization, also known as factoring, is the process of representing an integer or the other quantitative object as a product of several factors, which are generally smaller or easier objects that are the same.

We begin by attempting to find any rational roots using the Rational Root Theorem, which states that the possible rational roots are the positive or negative versions of the possible fractional combinations formed by placing a factor of the constant term in the numerator and a factor of the leading coefficient in the denominator.

The leading coefficient is the number in front of the largest power of the variable. When the terms are listed in descending order (highest to lowest power), the leading coefficient is always the first number. In our case the leading coefficient is hard to spot. Since there is no number in front of , the coefficient is 1 by default.

We then create all the possible fractions with a factor of the constant in the numerator and a factor of the leading coefficient in the denominator. This actually isn't as bad as it could be since our only possible denominator is 1. Any fraction with a denominator of 1 is just the numerator. Therfore, our possible "fractions" are simply

Unfortunately, this is where the process (at least without the assitance of a graphing calculator) becomes less fun. Using synthetic division, we must simply try each possible root until we have success. There's really no consistent rule to tell us where to start. Generally starting with the smaller whole numbers is best because the synthetic division is easier. Therefore, we could begin with then proceed to , etc.

Expansion and simplification give us which is properly expressed in standard form with the constant on the right. Below is a walkthrough of the expansion and simplification (including binomial and trinomial expansion techniques).

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