Algebraic Expressions

10x2+9x−4.

There is no factor that appears in each and every term. Hence, there are no common factors in this expression.

3x4+6x2+5x+8.

This expression has four terms: 3x4,6x2, 5x, and 8.

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).

14x5y+(a+3)2 contains two terms. Some of the factors of these terms are

first term:second term:14, x5, y(a+3) and (a+3)

9a2−6a−12 contains three terms. Some of the factors in each term are

first term:second term:third term:9 and a2, or, 9 and a and a−6 and a−12 and 1, or, 12 and −1

6(x2−y2)+19x(x2+y2)

no common factor

Exponents record the number of like factors in a term.

x . x . x . x = four factors = x raised 4

x2+5x2−9x2

x2

10(x−3) means there are ten (x−3)′s.

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12x means there are 12x′s.

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1y means there is one y. We usually write just y rather than 1y since it is clear just by looking that there is only one y.

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4ab means there are four ab′s.

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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.

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7a3 means there are seven a3′s.

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Coefficients

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Common Factors

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Identify the terms in the following expressions.

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List the terms in the following expressions.

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Name the common factors in each expression.

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Sample Set D

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x−3x2y/7+9x is an expression.

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In the expression 10+2(b+6)(b−18)2, list the factors of the first term: second term:

10 and 1 or 5 and 2; 2, b+6, b−18, b−18

14a2b2c(c−7)(2c+5)+28c(2c+5)

14c(2c+5)

In this expression there is only one term. The term is 15y8.

15y8.

4(a+1)3+10(a+1)

2(a+1)

2xy+6x2+(x−y)4

2xy, 6x2, (x−y)4

9ab(a−8)−15a(a−8)2

3a(a−8)

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.

4x2−8x3+16x4−24x5

4x2

4x2−8x+7

4x2, −8x, 7

5x2+3x−3xy7+(x−y)(x3−6)

5x2,3x,−3xy7, (x−y)(x3−6)

This example shows us that it is important for us to be very clear as to which quantity we are working with.

6x2y9 means there are six x2y9′s. It could also mean there are 6x2y9′s. It could even mean there are 6y9x2′s.

x+4 is an expression.

7y is an expression.

In the expression 8x2−5x+6, list the factors of the first term: second term: third term:

8, x, x; −5, x; 6 and 1 or 3 and 2

x+x+x+x = 4 terms

=4x here coefficients is 4

a number, a letter, or a collection of numbers and letters along with meaningful signs of operation.

Algebraic Expressions

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.

It is important to keep in mind the difference between coefficients and exponents.

Coefficients record the number of like terms in an algebraic expression.

where 4 is a exponent

In a term, the coefficient of a particular group of factors is the remaining group of factors.

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.

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.

5ax means there are five ax′s.

It could also mean there are 5ax′s.

Factors are parts of products and are therefore joined by multiplication signs.

Sample Set A

CoefficientIn algebra, as we now know, a letter is often used to represent some quantity.

Suppose we represent some quantity by the letter x.

FactorsAny 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.

Terms and Factors

TermsIn an algebraic expression, the quantities joined by "+" signs are called terms.

What does the coefficient of a quantity tell us?

The Difference Between Coefficients and Exponents

9(4−a).

The coefficient of (4−a) is 9.

4(x+y)2.

The coefficient of (x+y)2 is 4;

Often, the numerical coefficient is just called the coefficient.

The coefficient of a quantity records how many of that quantity there are.

6a3.

The coefficient of a3 is 6.

3x.

The coefficient of x is 3.

3x2y.

The coefficient of x2y is 3; the coefficient of y is 3x2; and the coefficient of 3 is x2y.

3/8xy4.

The coefficient of xy4 is 3/8.

3(x+5)−8(x+5).

The factor (x+5) appears in each term. So, (x+5) 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.

5x3−7x3+14x3.

The factor x3 appears in each and every term. The expression x3 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 a factor (since each of the individual quantities is joined by a multiplication sign). Thus, 5x2(x−7) 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.)