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
\(\dfrac{x-3x^2y}{7+9x}\) 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 \(8 \cdot 1\).
\(3x^2 + 6 = 4x - 1\) 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.
\(3x^4 + 6x^2 + 5x + 8\)
The expression has four terms: \(3x^4, 6x^2, 5x, 8\).
\(15y^8\)
In this expression there is only one term. The term is \(15y^8\).
\(14x^5y + (a+3)^2\).
In this expression there are two terms: the terms are \(14x^5y\) 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 \(3\).
\(m^3 - 3\)
Using our definition of subtraction, this expression can be written in the form \(m^3+(−3)\). Now we can see that the terms are \(m^3\) 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.
\(p^4-7p^3-2p-11\).
Associating the sign with the individual quantities we see that the terms of this expression are \(p^4\), \(−7p^3\), \(−2p\), 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
\(4x^2 - 8x + 7\)
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\(4x^2, -8x, 7\)
\(2xy + 6x^2 + (x-y)^4\)
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\(2xy, 6x^2, (x-y)^4\)
\(5x^2 + 3x - 3xy^7 + (x-y)(x^3-6)\)
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\(5x^2, 3x, -3xy^7, (x-y)(x^3-6)\)
Sample Set B
Identify the factors in each term.
\(9a^2 - 6a - 12\) contains three terms. Some of the factors in each term are
first term: \(9\) and \(a^2\), or \(9\) and \(a\) and \(a\)
second term: \(-6\) and \(a\)
third term: \(−12\) and \(1\), or, \(12\) and \(−1\)
\(14x^5y + (a+3)^2\) contains two terms. Some of the factors of these terms are
first term: \(14, x^5, y\)
second term: \((a+3)\) and \((a+3)\)
Practice Set B
In the expression \(8x^2 - 5x + 6\), list the factors of the:
first term:
second term:
third term:
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\(8, x, x\)
\(-5, x\)
\(6 \text{ and } 1\) or \(3 \text { and } 2\)
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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\(10 \text{ and } 1\) or \(5 \text{ and } 2\)
\(2, b+6, b−18, b−18\)
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.
\(5x^3 - 7x^3 + 14x^3\)
The factor x^3 appears in each and every term. The expression x^3 is a common factor.
\(4x^2 + 7x\)
The factor \(x\) appears in each term. The term \(4x^2\) is actually \(4xx\). Thus, \(x\) is a common factor.
\(12xy^2 - 9xy + 15\)
The only factor common to all three terms is the number 3. (Notice that \(12=3\cdot4, 9=3\cdot3, 15=3\cdot5\).
\(3(x+5) - 8(x+5)\).
The factor \((x+5)\) appears in each term. So, \((x+5)\) is a common factor.
\(45x^3(x-7)^2 + 15x^2(x-7) - 20x^2(x-7)^5\).
The number \(5\), the \(x^2\), and the \((x-7)\) appear in each term. Also, \(5x^2(x-7)\) is also a factor (since each of the individual quantities is joined by a multiplication sign). Thus, \(5x^2(x-7)\) is a common factor.
\(10x^2+9x-4\)
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.
\(x^2 + 5x^2 - 9x^2\)
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\(x^2\)
\(4x^2 - 8x^3 + 16x^4 - 24x^5\)
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\(4x^2\)
\(4(a+1)^3 + 10(a+1)\)
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\(2(a+1)\)
\(9ab(a-8) - 15a(a-8)^2\)
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\(3a(a-8)\)
\(14a^2b^2c(c-7)(2c+5) + 28c(2c+5)\)
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\(14c(2c+5)\)
\(6(x^2-y^2) + 19x(x^2+y^2)\)
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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 \(y\).
\(7a^3\) means there are seven \(a^3\)'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 \(x\)?".
\(6x^2y^9\) means there are six \(x^2y^9\)'s. It could also mean there are \(6x^2y^9\)'s. It could even mean there are \(6y^9x^2\)'s.
\(5x^3(y-7)\) means there are five \(x^3(y-7)\)'s. It could also mean there are \(5x^3(x-7)\)'s. It could also mean there are \(5(x-7)x^3\)'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. \(\underbrace{x+x+x+x}_{\text {4 terms }}=\underbrace{4 x}_{\text {coefficient is } 4}\)
Exponents record the number of like factors in a term. \(\underbrace{x \cdot x \cdot x \cdot x}_{\text {4 factors }}=\underbrace{x^{4}}_{\text { exponent is } 4}\)
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 \(3\).
\(6a^3\)
The coefficient of \(a^3\) is \(6\).
\(9(4-a)\)
The coefficient of \((4-a)\) is \(9\).
\(\dfrac{3}{8}xy^4\).
The coefficient of \(xy^4\) is \(\dfrac{3}{8}\)
\(3x^2y\).
The coefficient of \(x^2y\) is \(3\); the coefficient of \(y\) is \(3x^2\); and the coefficient of \(3\) is \(x^2y\).
\(4(x+y)^2\).
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 \(6x^3\), the coefficient of:
(a) \(x^3\) is.
(b) \(6\) is.
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(a) \(6\)
(b) \(x^3\)
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 \(10ab^4\), the coefficient of
(a) \(ab^4\) is .
(b) \(b^4\) is .
(c) \(a\) is .
(d) \(10\) is .
(e) \(10ab^3\) is .
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(a) \(10\)
(b) \(10a\)
(c) \(10b^4\)
(d) \(ab^4\)
(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
\(6x\), \(-10\)
\(2x^3+x-15\)
\(5x^2+6x-2\)
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three
\(5x^2, 6x, -2\)
\(3x\)
\(5cz\)
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one
\(5cz\)
\(2\)
\(61\)
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one
\(61\)
\(x\)
\(4y^3\)
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one
\(4y^3\)
\(17ab^2\)
\(a+1\)
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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.
\(x^2 + 5x^2 - 2x^2\)
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\(x^2\)
\(11y^3 - 33y^3\)
\(45ab^2 + 9b^2\)
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\(9b^2\)
\(6x^2y^3 + 18x^2\)
\(2(a+b) - 3(a+b)\)
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\((a+b)\)
\(14ab^2c^2(c+8) + 12ab^2c^2\)
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\(2ab^2c^2\)
\(4x^2y + 5a^2b\)
\(9a(a-3)^2 + 10b(a-3)\)
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\((a-3)\)
\(15x^2 - 30xy^2\)
\(12a^3b^2c - 7(b+1)(c-a)\)
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no common factors
\(0.06ab^2 + 0.03a\)
\(5.2(a+7)^2+17.1(a+7)\)
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\((a+7)\)
\(\dfrac{3}{4}x^2y^2z^2 + \dfrac{3}{8}x^2z^2\)
\(\dfrac{9}{16}(a^2-b^2) + \dfrac{9}{32}(b^2-a^2)\)
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\(\dfrac{9}{32}\)
For the following problems, note how many:
\(a\)'s in \(4a\)?
\(z\)'s in \(12z\)?
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\(12\)
\(x^2\)'s in \(5x^2\)?
\(y^3\)'s in \(6y^3\)?
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\(6\)
\(xy\)'s in \(9xy\)?
\(a^2b\)'s in \(10a^2b\)?
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\(10\)
\((a+1)\)'s in \(4(a+1)\)?
\((9+y)\)'s in \(8(9+y)\)?
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\(8\)
\(y^2\)'s in \(3x^3y^2\)?
\(12x\)'s in \(12x^2y^5\)?
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\(xy^5\)
\((a+5)\)'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 \(2x^2(x-7)\)?
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\(x^2(x-7)\)
\(3(a+8)\)'s in \(6x^2(a+8)^3(a-8)\)?
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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\(a\)
\(12a^2b^3c^2r^7; a^2c^2r^7\)
\(6x^2b^2(c-1); c-1\)
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\(6x^2b^2\)
\(10x(x+7)^2; 10(x+7)\)
\(9a^2b^5; 3ab^3\)
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\(3ab\)
\(15x^4y^4(z+9a)^3; (z+9a)\)
\((-4)a^6b^2; ab\)
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\((-4)a^5b\)
\((-11a)(a+8)^3(a-1); (a+8)^2\)
Exercises for Review
Simplify \([\dfrac{2x^8(x-1)^5}{x^4(x-1)^2}]^4\)
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\(16x^{16}(x-1)^{12}\)
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 \(\dfrac{2^5-4^2}{3^{-2}}\)
Express 0.0000152 using scientific notation.
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\(1.52 \times 10^{-5}\)