If we desire to multiply a sum by another number, one of two people we can multiply each term of the amount by the number prior to we include or we can first add the terms and then multiply. For example,


In either instance the result is the same.

You are watching: Factor x^2+x-12

This property, i m sorry we an initial introduced in section 1.8, is called the distributive law. In symbols,

a(b + c) = abdominal + ac or (b + c)a = ba + ca

By applying the distributive law to algebraic expressions containing parentheses, us can achieve equivalent expressions there is no parentheses.

Our an initial example requires the product the a monomial and also binomial.

Example 1 compose 2x(x - 3) without parentheses.


We think the 2x(x - 3) together 2x and then use the distributive regulation to obtain


The above technique works equally also with the product of a monomial and also trinomial.

Example 2 create - y(y2 + 3y - 4) there is no parentheses.


Applying the distributive property yields


When simplifying expressions including parentheses, we first remove the parentheses and also then incorporate like terms.

Example 3 simplify a(3 - a) - 2(a + a2).

We begin by remove parentheses come obtain


Now, combining like terms yields a - 3a2.

We can use the distributive building to rewrite expressions in i beg your pardon the coefficient of one expression in clip is +1 or - 1.

Example 4 create each expression there is no parentheses.a. +(3a - 2b)b. -(2a - 3b)



Notice that in instance 4b, the authorize of each term is readjusted when the expression is composed without parentheses. This is the same result that we would have obtained if we used the steps that we introduced in ar 2.5 to leveling expressions.


From the symmetric residential property of equality, we recognize that if

a(b + c) = ab + ac, then abdominal muscle + ac = a(b + c)

Thus, if there is a monomial factor typical to every terms in a polynomial, we have the right to write the polynomial as the product the the usual factor and another polynomial. For instance, due to the fact that each hatchet in x2 + 3x consists of x together a factor, we have the right to write the expression together the product x(x + 3). Rewriting a polynomial in this means is referred to as factoring, and the number x is claimed to be factored "from" or "out of" the polynomial x2 + 3x.

To element a monomial native a polynomial:Write a collection of parentheses came before by the monomial common to every term in the polynomial.Divide the monomial element into each term in the polynomial and write the quotient in the parentheses.Generally, we can discover the usual monomial variable by inspection.

Example 1 a. 4x + 4y = 4(x + y) b. 3xy -6y - 3y(x - 2)

We can examine that we factored properly by multiplying the factors and verifyingthat the product is the original polynomial. Using example 1, us get


If the typical monomial is tough to find, we can write every term in prime factored type and note the common factors.

Example 2 variable 4x3 - 6x2 + 2x.

solution We have the right to write


We now see that 2x is a typical monomial factor to all three terms. Climate we element 2x the end of the polynomial, and also write 2x()

Now, we division each term in the polynomial by 2x


and compose the quotients inside the parentheses to get

2x(2x2 - 3x + 1)

We can check our price in example 2 by multiplying the components to obtain


In this book, we will certainly restrict the common factors to monomials consisting of number coefficients that are integers and also to integral strength of the variables. The an option of authorize for the monomial variable is a issue of convenience. Thus,

-3x2 - 6x

can be factored either as

-3x(x + 2) or together 3x(-x - 2)

The first form is usually an ext convenient.

Example 3Factor out the typical monomial, consisting of -1.

a. - 3x2 - 3 xyb. -x3 - x2 + x solution


Sometimes it is practically to compose formulas in factored form.

Example 4 a. A = ns + PRT = P(1 + RT) b. S = 4kR2 - 4kr2 = 4k(R2 - r2)

4.3BINOMIAL products I

We have the right to use the distributive law to multiply 2 binomials. Although there is tiny need to multiply binomials in arithmetic as presented in the instance below, the distributive law likewise applies to expressions containing variables.


We will certainly now apply the over procedure because that an expression containing variables.

Example 1

Write (x - 2)(x + 3) without parentheses.

Solution First, apply the distributive residential property to get


Now, combine like state to achieve x2 + x - 6

With practice, girlfriend will be able to mentally add the 2nd and third products. Theabove procedure is sometimes referred to as the silver paper method. F, O, I, and also L stand for: 1.The product of the very first terms.2.The product the the outer terms.3.The product that the inner terms.4.The product the the last terms.

The FOIL an approach can likewise be offered to square binomials.

Example 2

Write (x + 3)2 without parentheses.Solution

First, rewrite (x + 3)2 as (x + 3)(x + 3). Next, use the FOIL technique to get


Combining choose terms yieldsx2 + 6x + 9

When we have a monomial factor and also two binomial factors, it is easiest to an initial multiply the binomials.

Example 3

create 3x(x - 2)(x + 3) without parentheses.Solution First, multiply the binomials come obtain3x(x2 + 3x - 2x - 6) = 3x(x2 + x - 6)

Now, apply the distributive regulation to gain 3x(x2 + x - 6) = 3x3 + 3x2 - 18x

Common Errors

Notice in instance 2




In general,



In section 4.3, we saw how to uncover the product of 2 binomials. Currently we will reverse this process. That is, offered the product of two binomials, us will discover the binomial factors. The process involved is one more example that factoring. As before,we will only consider factors in which the terms have integral number coefficients. Such determinants do not always exist, yet we will research the situations where they do.

Consider the adhering to product.


Notice that the very first term in the trinomial, x2, is product (1); the critical term in thetrinomial, 12, is product and the center term in the trinomial, 7x, is the amount of products (2) and (3).In general,


We usage this equation (from right to left) come factor any trinomial of the type x2 + Bx + C. We discover two numbers whose product is C and also whose sum is B.

Example 1 factor x2 + 7x + 12.Solution we look for two integers whose product is 12 and whose amount is 7. Think about the following pairs of components whose product is 12.


We see that the just pair of factors whose product is 12 and also whose amount is 7 is 3 and also 4. Thus,

x2 + 7x + 12 = (x + 3)(x + 4)

Note that when all terms of a trinomial space positive, we need only think about pairs of positive factors because we are in search of a pair of components whose product and also sum are positive. That is, the factored ax of

x2 + 7x + 12would it is in of the kind

( + )( + )

When the first and 3rd terms that a trinomial space positive yet the center term is negative, we need only consider pairs of an unfavorable factors since we are looking for a pair of determinants whose product is positive yet whose amount is negative. The is,the factored type of

x2 - 5x + 6

would be of the form


Example 2 aspect x2 - 5x + 6.

Solution since the third term is positive and the middle term is negative, we uncover two an unfavorable integers whose product is 6 and also whose sum is -5. We list the possibilities.


We check out that the just pair of determinants whose product is 6 and whose amount is -5 is -3 and also -2. Thus,

x2 - 5x + 6 = (x - 3)(x - 2)

When the an initial term the a trinomial is positive and the 3rd term is negative,the indicators in the factored type are opposite. That is, the factored form of

x2 - x - 12

would it is in of the form

(+)(-) or (-)(+)

Example 3

Factor x2 - x - 12.

Solution we must find two integers who product is -12 and also whose amount is -1. We list the possibilities.


We see that the only pair of factors whose product is -12 and whose sum is -1 is -4 and 3. Thus,

x2 - x - 12 = (x - 4)(x + 3)

It is easier to variable a trinomial totally if any monimial factor common to each term that the trinomial is factored first. For example, we can factor

12x2 + 36x + 24



A monomial can then it is in factored from this binomial factors. However, first factoring the common factor 12 native the original expression yields

12(x2 + 3x + 2)

Factoring again, we have

12(* + 2)(x + 1)

which is stated to it is in in fully factored form. In such cases, it is not essential to factor the numerical variable itself, that is, we execute not create 12 together 2 * 2 * 3.

instance 4

aspect 3x2 + 12x + 12 completely.

SolutionFirst we aspect out the 3 indigenous the trinomial to gain

3(x2 + 4x + 4)

Now, we factor the trinomial and obtain

3(x + 2)(x + 2)

The techniques we have developed are additionally valid for a trinomial such together x2 + 5xy + 6y2.

Example 5Factor x2 + 5xy + 6y2.

Solution We discover two positive factors whose product is 6y2 and also whose sum is 5y (the coefficient that x). The two factors are 3y and 2y. Thus,

x2 + 5xy + 6y2 = (x + 3y)(x + 2y)

as soon as factoring, the is ideal to create the trinomial in descending powers of x. If the coefficient the the x2-term is negative, variable out a negative before proceeding.

Example 6

Factor 8 + 2x - x2.

Solution We first rewrite the trinomial in descending strength of x come get

-x2 + 2x + 8

Now, us can aspect out the -1 come obtain

-(x2 - 2x - 8)

Finally, we element the trinomial come yield

-(x- 4)(x + 2)

Sometimes, trinomials space not factorable.

Example 7

Factor x2 + 5x + 12.

Solution we look for 2 integers who product is 12 and whose amount is 5. From the table in instance 1 on web page 149, we view that over there is no pair of factors whose product is 12 and also whose sum is 5. In this case, the trinomial is no factorable.

Skill at factoring is usually the an outcome of substantial practice. If possible, do the factoring process mentally, writing your prize directly. You can check the outcomes of a administer by multiply the binomial factors and also verifying the the product is equal to the offered trinomial.

4.5BINOMIAL assets II

In this section, we usage the procedure occurred in ar 4.3 to main point binomial determinants whose first-degree terms have numerical coefficients various other than 1 or - 1.

Example 1

Write as a polynomial.

a. (2x - 3)(x + 1)b. (3x - 2y)(3x + y)


We first apply the FOIL technique and then incorporate like terms.


As before, if we have actually a squared binomial, we first rewrite it together a product, then use the silver paper method.

Example 2

a. (3x + 2)2 = (3x + 2)(3x + 2) = 9x2 + 6x + 6x + 4 = 9x2 + 12x + 4

b. (2x - y)2 = (2x - y)(2x - y) = 4x2 - 2xy - 2xy + y2 - 4x2 - 4xy + y2

As you may have actually seen in ar 4.3, the product of 2 bionimals may have actually no first-degree hatchet in the answer.

Example 3

a. (2x - 3)(2x + 3) = 4x2 + 6x - 6x - 9 = 4x2 -9

b. (3x - y)(3x + y) - 9x2 + 3xy - 3xy - y2= 9x2 - y2

When a monomial factor and also two binomial determinants are gift multiplied, that iseasiest to main point the binomials first.

Example 4

Write 3x(2x - l)(x + 2) together a polynomial.

Solution We first multiply the binomials to get3x(2x2 + 4x - x - 2) = 3x(2x2 + 3x - 2)Now multiply by the monomial yields3x(2x2) + 3x(3x) + 3x(-2) = 6x3 + 9x2 - 6x


In ar 4.4 we factored trinomials that the type x2 + Bx + C wherein the second-degree term had a coefficient of 1. Currently we desire to extend our factoring techniquesto trinomials that the form Ax2 + Bx + C, wherein the second-degree term has actually acoefficient various other than 1 or -1.

First, we think about a check to identify if a trinomial is factorable. A trinomial ofthe kind Ax2 + Bx + C is factorable if we can find two integers whose product isA * C and whose amount is B.

Example 1

Determine if 4x2 + 8x + 3 is factorable.

Solution We inspect to check out if there space two integers whose product is (4)(3) = 12 and whosesum is 8 (the coefficient of x). Think about the complying with possibilities.


Since the components 6 and 2 have actually a amount of 8, the value of B in the trinomialAx2 + Bx + C, the trinomial is factorable.

Example 2

The trinomial 4x2 - 5x + 3 is no factorable, because the above table shows thatthere is no pair of factors whose product is 12 and also whose sum is -5. The check tosee if the trinomial is factorable have the right to usually be done mentally.

Once us have figured out that a trinomial that the type Ax2 + Bx + C is fac-torable, we proceed to uncover a pair of components whose product is A, a pair the factorswhose product is C, and an setup that yields the suitable middle term. Weillustrate by examples.

Example 3

Factor 4x2 + 8x + 3.

Solution Above, we determined that this polynomial is factorable. We currently proceed.

1. We think about all pairs of components whose product is 4. Because 4 is positive, just positive integers should be considered. The possibilities are 4, 1 and also 2, 2.2. We take into consideration all bag of determinants whose product is 3. Because the middle term is positive, think about positive pairs of factors only. The possibilities room 3, 1. We create all feasible arrangements the the components as shown.

3. We pick the plan in i m sorry the amount of products (2) and also (3) returns a center term the 8x.

Now, we take into consideration the administrate of a trinomial in which the constant term is negative.

Example 4

Factor 6x2 + x - 2.

Solution First, us test to see if 6x2 + x - 2 is factorable. Us look for 2 integers that havea product that 6(-2) = -12 and a amount of 1 (the coefficient that x). The integers 4 and-3 have actually a product of -12 and a amount of 1, for this reason the trinomial is factorable. We nowproceed.

We think about all pairs of determinants whose product is 6. Due to the fact that 6 is positive, only positive integers have to be considered. Then possibilities room 6, 1 and also 2, 3.We consider all pairs of components whose product is -2. The possibilities are 2, -1 and -2, 1. We compose all possible arrange ments the the components as shown.We select the plan in which the sum of products (2) and also (3) returns a middle term the x.

With practice, friend will be able to mentally examine the combinations and will notneed to create out all the possibilities. Paying fist to the indicators in the trinomialis specifically helpful because that mentally eliminating possible combinations.

It is most basic to element a trinomial written in descending strength of the variable.

Example 5


a. 3 + 4x2 + 8x b. X - 2 + 6x2

Solution Rewrite each trinomial in descending powers of x and then monitor the solutions ofExamples 3 and 4.

a. 4x2 + 8x + 3 b. 6x2 + x - 2

As we claimed in ar 4.4, if a polynomial contains a common monomial factorin each of the terms, us should factor this monomial indigenous the polynomial beforelooking for other factors.

Example 6

Factor 242 - 44x - 40.

Solution We very first factor 4 from every term come get

4(6x2 - 11x - 10)

We then aspect the trinomial, to obtain

4(3x + 2)(2x - 5)


If the over "trial and error" technique of factoring does no yield rapid results, analternative method, i m sorry we will certainly now demonstrate using the earlier example4x2 + 8x + 3, may be helpful.

We understand that the trinomial is factorable because we uncovered two number whoseproduct is 12 and also whose amount is 8. Those numbers room 2 and also 6. We currently proceedand usage these numbers to rewrite 8x as 2x + 6x.

We now factor the an initial two terms, 4*2 + 2x and the last 2 terms, 6x + 3.A typical factor, 2x + 1, is in every term, so us can element again.This is the same result that we obtained before.


Some polynomials happen so frequently that the is useful to identify these specialforms, i beg your pardon in tum allows us to directly write your factored form. Observe that


In this section we room interested in viewing this connection from right to left, fromthe polynomial a2 - b2 come its factored kind (a + b)(a - b).

The distinction of 2 squares, a2 - b2, equates to the product of the sum a + b and the distinction a - b.

Example 1

a. X2 - 9 = x2 - 32 = (x + 3)(x - 3) b. X2 - 16 = x2 - 42 = (x + 4)(x - 4)


(3x)(3x) = 9x2

we deserve to view a binomial such as 9x2 - 4 together (3x)2 - 22 and also use the above methodto factor.

Example 2

a.9x2 - 4 = (3x)2 - 22= (3x + 2)(3x - 2)b.4y2 - 25x2 = (2y)2 - (5x)2= (2y + 5x)(2y - 5x)

As before, we always factor out a common monomial first whenever possible.

Example 3

a.x3 - x5 = x3(l - x2) = x3(1 + x)(l - x)b.a2x2y - 16y = y(a2x2 - 16) = y<(ax)2 - 42>= y(ax - 4 )(ax + 4)


Often we have to solve equations in i m sorry the change occurs within parentheses. Wecan settle these equations in the usual manner ~ we have actually simplified lock byapplying the distributive law to eliminate the parentheses.

Example 1

Solve 4(5 - y) + 3(2y - 1) = 3.

Solution We first apply the distributive law to get

20 - 4y + 6y - 3 = 3

Now combining prefer terms and also solving because that y yields

2y + 17 = 3

2y = -14


The same method can be applied to equations including binomial products.

Example 2

Solve (x + 5)(x + 3) - x = x2 + 1.

Solution First, we apply the FOIL an approach to remove parentheses and also obtain

x2 + 8x + 15 - x = x2 + 1

Now, combining choose terms and also solving because that x yields

x2 + 7x + 15 = x2 + 1

7x = -14

x = -2


Parentheses are valuable in representing assets in which the change is containedin one or much more terms in any type of factor.

Example 1

One integer is three more than another. If x to represent the smaller integer, representin regards to x

a. The larger integer.b. 5 times the smaller sized integer.c. Five times the bigger integer.

Solution a. X + 3b. 5x c. 5(x + 3)

Let us say we understand the amount of two numbers is 10. If we represent one number byx, climate the second number need to be 10 - x as suggested by the following table.


In general, if we understand the sum of two numbers is 5 and x to represent one number,the various other number must be S - x.

Example 2

The amount of 2 integers is 13. If x to represent the smaller sized integer, stand for in termsof X

a. The bigger integer.b. Five times the smaller integer.c. 5 times the larger integer.

Solution a. 13 - x b. 5x c. 5(13 - x)

The next example comes to the id of consecutive integers the was consid-ered in section 3.8.

Example 3

The distinction of the squares of two consecutive odd integers is 24. If x representsthe smaller sized integer, represent in terms of x

a. The larger integerb. The square that the smaller sized integer c. The square of the larger integer.


a. X + 2b. X2 c. (x + 2)2

Sometimes, the math models (equations) because that word problems involveparentheses. We have the right to use the approach outlined on web page 115 to obtain the equation.Then, we continue to fix the equation by first writing equivalently the equationwithout parentheses.

Example 4

One creature is five more than a second integer. Three times the smaller integer plustwice the larger amounts to 45. Discover the integers.


Steps 1-2 First, we create what we want to uncover (the integers) together word phrases. Then, we stand for the integers in regards to a variable.The smaller integer: x The bigger integer: x + 5

Step 3 A sketch is not applicable.

Step 4 Now, we create an equation that represents the condition in the problemand get

3x + 2(x + 5) = 45

Step 5 applying the distributive regulation to remove parentheses yields


Step 6 The integers space 7 and 7 + 5 or 12.


In this section, we will examine several applications of word problems that lead toequations the involve parentheses. When again, we will certainly follow the six procedures out-lined on web page 115 when we fix the problems.


The straightforward idea of difficulties involving coins (or bills) is the the worth of a numberof coins that the very same denomination is same to the product the the worth of a singlecoin and the total variety of coins.


A table choose the one shown in the next example is useful in resolving coin problems.

Example 1

A collection of coins consisting of dimes and quarters has actually a worth of $5.80. Thereare 16 an ext dimes 보다 quarters. How countless dimes and also quarters room in the col-lection?


Steps 1-2 We first write what we want to find as indigenous phrases. Then, werepresent each phrase in terms of a variable.The number of quarters: x The variety of dimes: x + 16

Step 3 Next, us make a table showing the number of coins and also their value.


Step 4 currently we can write an equation.


Step 5 fixing the equation yields


Step 6 There room 12 quarters and also 12 + 16 or 28 dimes in the collection.


The an easy idea of fixing interest difficulties is the the quantity of interest i earnedin one year at an easy interest equates to the product the the price of attention r and theamount of money ns invested (i = r * p). For example, $1000 invested because that one yearat 9% returns i = (0.09)(1000) = $90.

A table prefer the one shown in the next instance is valuable in fixing interestproblems.

Example 2

Two investments produce an annual interest that $320. $1000 much more is invested at11% 보다 at 10%. Exactly how much is invest at every rate?


Steps 1-2 We very first write what we desire to discover as word phrases. Then, werepresent each phrase in terms of a variable. Amount invest at 10%: x Amount invest at 11%: x + 100

Step 3 Next, us make a table showing the lot of money invested, therates the interest, and the quantities of interest.


Step 4 Now, we deserve to write an equation relating the attention from every in-vestment and also the full interest received.


Step 5 To resolve for x, an initial multiply every member by 100 come obtain


Step 6 $1000 is invest at 10%; $1000 + $1000, or $2000, is invested at11%.


The an easy idea of fixing mixture difficulties is that the lot (or value) that thesubstances being mixed must same the amount (or value) the the last mixture.

A table prefer the ones displayed in the following instances is beneficial in solvingmixture problems.

Example 3

How lot candy worth 80c a kilogram (kg) have to a grocer blend v 60 kg ofcandy worth $1 a kilogram to do a mixture precious 900 a kilogram?


Steps 1-2 We first write what we want to discover as a indigenous phrase. Then, werepresent the expression in terms of a variable.Kilograms that 80c candy: x

Step 3 Next, we make a table showing the varieties of candy, the amount of each,and the complete values of each.


Step 4 We have the right to now write an equation.


Step 5 addressing the equation yields


Step 6 The grocer must use 60 kg of the 800 candy.

Another type of mixture trouble is one that requires the mixture the the two liquids.

Example 4

How many quarts the a 20% solution of acid have to be added to 10 quarts the a 30%solution of acid to achieve a 25% solution?


Steps 1-2 We first write what we want to uncover as a word phrase. Then, werepresent the phrase in terms of a variable.

Number the quarts that 20% equipment to it is in added: x

Step 3 Next, us make a table or illustration showing the percent of each solu-tion, the quantity of each solution, and also the amount of pure acid in eachsolution.



Step 4 We can now create an equation relating the quantities of pure mountain beforeand after combine the solutions.


Step 5 To settle for x, an initial multiply each member by 100 come obtain

20x + 30(10) = 25(x + 10)20x + 300 = 25x + 250 50 = 5x 10 = x

Step 6 add 10 quarts of 20% solution to produce the wanted solution.


Algebraic expression containing parentheses can be created without bracket byapplying the distributive law in the forma(b + c) = abdominal + ac

A polynomial that consists of a monomial factor typical to all terms in thepolynomial have the right to be created as the product the the usual factor and also anotherpolynomial by using the distributive legislation in the formab + ac = a(b + c)

The distributive law can be offered to multiply binomials; the FOIL technique suggeststhe four assets involved.


Given a trinomial the the form x2 + Bx + C, if there space two numbers, a and b,whose product is C and whose sum is B, climate x2 + Bx + C = (x + a)(x + b) otherwise, the trinomial is no factorable.

A trinomial that the kind Ax2 + Bx + C is factorable if there are two numbers whoseproduct is A * C and also whose amount is B.

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The distinction of squaresa2 - b2 = (a + b)(a - b)

Equations involving parentheses can be fixed in the usual means after the equationhas to be rewritten equivalently there is no parentheses.