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A L G E B R A

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THE PARTS OF AN EXPRESSION THAT ARE BEING ADDED OR SUBTRACTED are called its terms. When the factors of a term are a number and a letter or letters, then the number is called the coefficient of the other factor.

Thus in the product 5ab, the factor 5 is the coefficient of ab.

Terms that do not differ at all, except for their coefficients, are called like terms. Here, for example, is a sum of like terms:

5ab + 3ab − 2ab.

We can always add or combine like terms by adding their coefficients. Upon adding 5 + 3 − 2:

5ab + 3ab − 2ab = 6ab.

We saw in Lesson 3 that we include the subtraction sign as part of the name of the term.

Adding like terms has the effect of reducing the number of terms, which is what we like.

Here, on the other hand, is a sum of unlike terms:

x² − 4xy + y²

There is no way that we could combine them.

The most elementary terms that we could combine are a and −a.

5 + (−5) = 0.

Every term has a coefficient. In this sum,

xy + 3z

the coefficient of x is understood to be 1; for x = 1·x. Since −y = (−1)y, the coefficient of y is −1.  (Lesson 5.)

It is the style in algebra not to write the coefficients 1 or −1.

Example 1.   Add the like terms:

2x + 3y + 4x − 5y

Solution.   The like terms are 2x and 4x,  3y and −5y.  Upon adding their coefficients:

2x + 3y + 4x5y = 6x − 2y.

We say that there are two terms "in" x and two "in" y. The preposition "in" indicates the like terms.

See Problem 12.

Problem 1 .   6a − 3b + cd.

a)  What number is the coefficient of a ?

6

b)  What number is the coefficient of b ?  −3

c)  What number is the coefficient of c ?   1

d)  What number is the coefficient of d ?   −1

Actually, the coefficient of any factor is all the remaining factors. Thus in the product 4ab, the coefficient of a is 4b; the coefficient of 4a is b; and so on. In this product, x(x − 1), the coefficient of (x − 1) is x.

Problem 2.   In the expression 5ayx, name the coefficient of

a)  x  5ay          b)  y  5ax          c)  yx  5a

d)  5a  xy          e)  5  ayx

Problem 3.   In this product  2(x + y)z

a)  name each factor.   2, (x + y), z

b)  name the coefficient of z.   2(x + y)

c)  name the coefficient of (x + y).   2z

Problem 4.   What number is the coefficient of x?

 a) x2 12 Compare Lesson 6, Problem 7b.
 b) 3x 4 34 3x 4 = 34 · x Lesson 4.

Problem 5.   How do we add like terms?

 a)  6x + 2x = 8x b)  6x − 2x = 4x c)  5x + x = 6x d)  5x − x = 4x e)  −4x + 5x = x f)  4x − 5x = −x Again, we do not to write the coefficients 1 or −1. g)  −5x − 3x = −8x h)  −x − x = −2x

i)  −3x − 4 + 2x + 6  = −x + 2

j)  x − 2 − 4x − 5  = −3x − 7

k)  4x + y − 2x + y = 2x + 2y

l)  3xy − 8x + 2y  = −5x + y

m)  4x² − 5x² + x² = 0

a)   2a + 3b   These are not like terms.

b)   2a + 3b + 4a − 5ab  = 6a + 3b − 5ab.
Terms that you cannot combine, simply rewrite.

Problem 8.    Remove parentheses and add like terms.

 a)   (2a − 3b + c) + (5a − 6b + c) = 2a − 3b + c + 5a − 6b + c = 7a − 9b + 2c
 b)    (a + 2b + 4c − 3d) − (3a − 8b − 2c + d) = a + 2b + 4c − 3d −3a + 8b + 2c − d = −2a + 10b + 6c − 4d
 c)    (4x − 3y) + (3y − 5x) + (5z − 4x) = 4x − 3y + 3y − 5x + 5z − 4x = −5x + 5z
 d)    (5xy − 3x + 2y − 1) − (2xy − 7x − 8y + 6) = 5xy − 3x + 2y − 1 −2xy + 7x + 8y − 6 = 3xy + 4x + 10y − 7
 e)   (x − y) − (y + xy − x) − (2x − 4xy − 2y) = x − y −y − xy + x − 2x + 4xy + 2y = 3xy
 f)   (4x² − 7x − 3) − (x² − 4x + 1) = 4x² − 7x − 3 − x² + 4x − 1 = 3x² − 3x − 4
 g)   (6x3 + 4x² − 2x − 6) − (2x3 − 8x² + x − 2) = 6x3 + 4x² − 2x − 6 − 2x3 + 8x² − x + 2 = 4x3 + 12x² − 3x − 4
 h)   (x² + x + 1) + (2x² + 2x + 2) − (x² − x − 1) = x² + x + 1 + 2x² + 2x + 2 − x² + x + 1 = 2x² + 4x + 4

Problem 9.   5abc + 2cba.  Are those like terms?

Yes.  The order of factors does not matter.
Upon combining them, we get 7abc.
When writing the final sum, it is conventional to preserve the alphabetical order.

 a) 4xy − 9yx  = −5xy b) 8x − 5xy − 4x + 4yx  = 4x − xy

c)   9xyz + 3yzx + 5zxy  = 17xyz

d)   3xy − 4xyz + 3x − 8yx + 5yzx − 9x  = −5xy + xyz − 6x

a)   2n + 2 − n =  n + 2

b)   n − 2 − 3n + 1 =  −2n − 1

c)   2n + 4 − 2n − 2 = 2

Problem 12.   Add like terms, which are in (x + 2). Do not remove parentheses.

a)  3(x + 2) + 7(x + 2) =  10(x + 2).

b)  2(x + 2) − 5(x + 2) =  −3(x + 2).

c)  x(x + 2) + 4(x + 2) =  (x + 4)(x + 2).

d)  x(x + 2) − (x + 2) =  (x − 1)(x + 2).

Problem 13.   Add like terms, which are in x or y. Add the coefficients.

a)   px + qx = (p + q)x.

b)   ax + bycx + dy = (ac)x + (b +d)y.

c)   x + ax = (1 + a)x.      d)   axx = (a − 1)x.

e)   (a + b)x + cx = (a + b + c)x.

f)   (ab)xcx = (abc)x.

f)   (a + b)x − (b + a)x = 0.

a)  3a2b3 − 2ab2 + a3b2 − 5b2a + b3a2 = 4a2b3 − 7ab2 + a3b2.

b)   xy2xy + x2yy2x + 2yx2 + yx = 3x2y.

*

In calculus, the student will not see any problem worded "Subtract a from b." However, in certain standard exams, that wording tends to come up. Hence, the following rule.

The rule for subtraction

"Subtract a from b."  Is that  ab  or  ba ?

It is  ba.   a is the number being subtracted.  It is called the subtrahend.  The subtrahend appears to the right of the minus sign -- before the word "from."

Example 2.   Subtract  2x − 3  from  5x − 4

Solution.   2x − 3 is the subtrahend.

 (5x − 4) − (2x − 3) = 5x − 4 − 2x + 3 = 3x − 1.

Notice:  The signs of the subtrahend change.

2x − 3  changes to  −2x + 3.

We can therefore state the following rule for subtraction.

Change the signs of all the terms in the subtrahend.

Problem 15.   Subtract  4a − 2b  from  a + 3b.

Change the signs of the subtrahend, and add:

a + 3b − 4a + 2b = −3a + 5b.

Problem 16.   Subtract  x² − 5x + 7  from  3x² − 8x − 2.

 3x² − 8x − 2 − x² + 5x − 7 = 2x² − 3x − 9. Next Lesson:  Linear equations

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