4.2: Algebraic Expressions
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Algebraic Expressions
An algebraic expression is a number, a letter, or a collection of numbers and letters along with meaningful signs of operation.
Algebraic expressions are often referred to simply as expressions, as in the following examples:
x+4 is an expression
7y is an expression
x−3x2y7+9x is an expression.
The number 8 is an expression. 8 can be written with explicit signs of operation by writing it as 8+0 or 8⋅1.
3x2+6=4x−1 is not an expression, it is an equation. We will study equations in the next section.
Terms and Factors
In an algebraic expression, the quantities joined by "+" signs are called terms.
In some expressions it will appear that terms are joined by "−" signs. We must keep in mind that subtraction is addition of the negative, that is a−b=a+(−b).
An important concept that all students of algebra must be aware of is the difference between terms and factors.
Any numbers or symbols that are multiplied together are factors of their product.
Terms are parts of sums and are therefore joined by addition (or subtraction) signs.
Factors are parts of products and are therefore joined by multiplication signs.
Sample Set A
Identify the terms in the following expressions.
3x4+6x2+5x+8
The expression has four terms: 3x4,6x2,5x,8.
15y8
In this expression there is only one term. The term is 15y8.
14x5y+(a+3)2.
In this expression there are two terms: the terms are 14x5y and (a+3)2. Notice that the term (a+3)2 is itself composed of two like factors, each of which is composed of the two terms, a and 3.
m3−3
Using our definition of subtraction, this expression can be written in the form m3+(−3). Now we can see that the terms are m3 and −3.
Rather than rewriting the expression when a subtraction occurs, we can identify terms more quickly by associating the + or − sign with the individual quantity.
p4−7p3−2p−11.
Associating the sign with the individual quantities we see that the terms of this expression are p4, −7p3, −2p, and −11.
Practice Set A
Let’s say it again. The difference between terms and factors is that terms are joined by signs and factors are joined by signs.
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addition, multiplication
4x2−8x+7
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4x2,−8x,7
2xy+6x2+(x−y)4
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2xy,6x2,(x−y)4
5x2+3x−3xy7+(x−y)(x3−6)
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5x2,3x,−3xy7,(x−y)(x3−6)
Sample Set B
Identify the factors in each term.
9a2−6a−12 contains three terms. Some of the factors in each term are
first term: 9 and a2, or 9 and a and a
second term: −6 and a
third term: −12 and 1, or, 12 and −1
14x5y+(a+3)2 contains two terms. Some of the factors of these terms are
first term: 14,x5,y
second term: (a+3) and (a+3)
Practice Set B
In the expression 8x2−5x+6, list the factors of the:
first term:
second term:
third term:
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8,x,x
−5,x
6 and 1 or 3 and 2
In the expression 10+2(b+6)(b−18)2, list the factors of the:
first term:
second term:
third term:
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10 and 1 or 5 and 2
2,b+6,b−18,b−18
Common Factors
Common Factors
Sometimes, when we observe an expression carefully, we will notice that some particular factor appears in every term. When we observe this, we say we are observing common factors. We use the phrase common factors since the particular factor we observe is common to all the terms in the expression. The factor appears in each and every term in the expression.
Sample Set C
Name the common factors in each expression.
5x3−7x3+14x3
The factor x^3 appears in each and every term. The expression x^3 is a common factor.
4x2+7x
The factor x appears in each term. The term 4x2 is actually 4xx. Thus, x is a common factor.
12xy2−9xy+15
The only factor common to all three terms is the number 3. (Notice that 12=3⋅4, 9=3⋅3, 15=3⋅5.
3(x+5)−8(x+5).
The factor (x+5) appears in each term. So, (x+5) is a common factor.
45x3(x−7)2+15x2(x−7)−20x2(x−7)5.
The number 5, the x2, and the (x−7) appear in each term. Also, 5x2(x−7) is also a factor (since each of the individual quantities is joined by a multiplication sign). Thus, 5x2(x−7) is a common factor.
10x2+9x−4
There is no factor that appears in each and every term. Hence, there are no common factors in this expression.
Practice Set C
List, if any appear, the common factors in the following expressions.
x2+5x2−9x2
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x2
4x2−8x3+16x4−24x5
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4x2
4(a+1)3+10(a+1)
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2(a+1)
9ab(a−8)−15a(a−8)2
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3a(a−8)
14a2b2c(c−7)(2c+5)+28c(2c+5)
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14c(2c+5)
6(x2−y2)+19x(x2+y2)
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No common factor.
Coefficients
Coefficient
In algebra, as we now know, a letter is often used to represent some quantity. Suppose we represent some quantity by the letter x. The notation 5x means x+x+x+x+x. We can now see that we have five of these quantities. In the expression 5x, the number 5 is called the numerical coefficient of the quantity x. Often, the numerical coefficient is just called the coefficient. The coefficient of a quantity records how many of that quantity there are.
Sample Set D
12x means that there are 12x's.
4ab means there are four ab's.
10(x−3) means there are ten x−3)'s.
1y means there is one y. We usually write just y rather than 1y since it is clear just be looking that there is only one y.
7a3 means there are seven a3's.
5ax means there are five ax's. It could also mean there are 5ax's. This example shows us that it is important for us to be very clear as to which quantity we are working with. When we see the expression 5ax we must ask ourselves "Are we working with the quantity ax or the quantity x?".
6x2y9 means there are six x2y9's. It could also mean there are 6x2y9's. It could even mean there are 6y9x2's.
5x3(y−7) means there are five x3(y−7)'s. It could also mean there are 5x3(x−7)'s. It could also mean there are 5(x−7)x3's.
Practice Set D
What does the coefficient of a quantity tell us?
The Difference Between Coefficients and Exponents It is important to keep in mind the difference between coefficients and exponents.
Coefficients record the number of like terms in an algebraic expression. x+x+x+x⏟4 terms =4x⏟coefficient is 4
Exponents record the number of like factors in a term. x⋅x⋅x⋅x⏟4 factors =x4⏟ exponent is 4
In a term, the coefficient of a particular group of factors is the remaining group of factors.
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How many of that quantity there are.
Sample Set E
3x
The coefficient of x is 3.
6a3
The coefficient of a3 is 6.
9(4−a)
The coefficient of (4−a) is 9.
38xy4.
The coefficient of xy4 is 38
3x2y.
The coefficient of x2y is 3; the coefficient of y is 3x2; and the coefficient of 3 is x2y.
4(x+y)2.
The coefficient of (x+y)2 is 4; the coefficient of 4 is (x+y)2; and the coefficient of (x+y) is 4(x+y) since 4(x+y)2 can be written as 4(x+y)(x+y).
Practice Set E
Determine the coefficients.
In the term 6x3, the coefficient of:
(a) x3 is.
(b) 6 is.
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(a) 6
(b) x3
In the term 3x(y−1), the coefficient of
(a) x(y−1) is.
(b) (y−1) is.
(c) 3(y−1) is.
(d) x is.
(e) 3 is .
(f) The numerical coefficient is .
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(a) 3
(b) 3x
(c) x
(d) 3(y−1)
(e) x(y−1)
(f) 3
In the term 10ab4, the coefficient of
(a) ab4 is .
(b) b4 is .
(c) a is .
(d) 10 is .
(e) 10ab3 is .
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(a) 10
(b) 10a
(c) 10b4
(d) ab4
(e) b
Exercises
For the expressions in the following problems, write the number of terms that appear and then list the terms.
What is an algebraic expression?
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An algebraic expression is a number, a letter, or a collection of numbers and letters along with meaningful signs of operation.
Why is the number 14 considered to be an expression?
Why is the number x considered to be an expression?
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x is an expression because it is a letter (see the definition).
For the following problems, list, if any should appear, the common factors in the expressions.
2x+1
6x−10
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two
6x, −10
2x3+x−15
5x2+6x−2
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three
5x2,6x,−2
3x
5cz
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one
5cz
2
61
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one
61
x
4y3
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one
4y3
17ab2
a+1
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two
a,1
(a+1)
2x+x+7
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three
2x,x,7
2x+(x+7)
(a+1)+(a−1)
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two
(a+b),(a−1)
a+1+(a−1)
For the following problems, list, if any should appear, the common factors in the expressions.
x2+5x2−2x2
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x2
11y3−33y3
45ab2+9b2
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9b2
6x2y3+18x2
2(a+b)−3(a+b)
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(a+b)
14ab2c2(c+8)+12ab2c2
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2ab2c2
4x2y+5a2b
9a(a−3)2+10b(a−3)
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(a−3)
15x2−30xy2
12a3b2c−7(b+1)(c−a)
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no common factors
0.06ab2+0.03a
5.2(a+7)2+17.1(a+7)
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(a+7)
34x2y2z2+38x2z2
916(a2−b2)+932(b2−a2)
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932
For the following problems, note how many:
a's in 4a?
z's in 12z?
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12
x2's in 5x2?
y3's in 6y3?
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6
xy's in 9xy?
a2b's in 10a2b?
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10
(a+1)'s in 4(a+1)?
(9+y)'s in 8(9+y)?
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8
y2's in 3x3y2?
12x's in 12x2y5?
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xy5
(a+5)'s in 2(a+5)?
(x−y)'s in 5x(x−y)?
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5x
(x+1)'s in 8(x+1)?
2's in 2x2(x−7)?
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x2(x−7)
3(a+8)'s in 6x2(a+8)3(a−8)?
For the following problems, a term will be given followed by a group of its factors. List the coefficient of the given group of factors.
7y;y
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7
10x;x
5a;5
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a
12a2b3c2r7;a2c2r7
6x2b2(c−1);c−1
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6x2b2
10x(x+7)2;10(x+7)
9a2b5;3ab3
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3ab
15x4y4(z+9a)3;(z+9a)
(−4)a6b2;ab
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(−4)a5b
(−11a)(a+8)3(a−1);(a+8)2
Exercises for Review
Simplify [2x8(x−1)5x4(x−1)2]4
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16x16(x−1)12
Supply the missing phrase. Absolute value speaks to the question of and not "which way."
Find the value of −[−6(−4−2)+7(−3+5)].
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−50
Find the value of 25−423−2
Express 0.0000152 using scientific notation.
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1.52×10−5