Terms in Algebraic Expressions - Grade 6

Grade 6 examples and questions on terms in algebraic expressions, with detailed solutions and explanations, are presented.


Algebraic Expression


Definition: An algebraic expression is made up of one or more terms and each term is either a signed number or a signed number multiplied by one or more variables raised to a certain power.
Example 1
\(2x + 4\) is called an algebraic expression
it has two terms \(2x\) and \(4\)
Example 2
\(-3x^{2} - x - 6\) is another expression and it has 3 terms: \(-3x^{2}\) , \(-x\) and \(-6\)
Example 3
\(-5xy - \frac{3}{4}x + y - 7\) is another expression and it has 4 terms: \(-5xy\), \(-\frac{3}{4}x\), \(y\) and \(-7\)
Note
Terms without variable such as \(4\), \(-6\) and \(-7\) are called constant terms.

Degree of a Term

Definition: The degree of a term is the sum of all powers of the variables in that term
a) The degree of the term \(2x\) is 1 because \(2x\) means \(2x^{1}\) and 1 is the power of \(x\).
b) The degree of the term \(-3x^{2}\) is 2 because 2 is the power of \(x\)
c) The degree of the term \(-5xy\) in example 3 is 2 because the power of \(x\) is 1 and the power of \(y\) is 1 and degree of the term is the sum of the two powers.
d) The degree of the term \(-x^{2}y\) is 3 because the power of \(x\) is 2 and the power of \(y\) is 1 and degree of the term is the sum of the two powers.
e) The degree of a constant term such as \(4\), \(-6\) and \(-7\) is zero.


Coefficient of a Term


Definition: The coefficient of a term is the signed number that multiplies the term
a) The coefficient of the term \(2x\) which can be written as \((2)x\) is 2
b) The coefficient of the term \(-3x^{2}\) which can be written as \((-3)x^{2}\) is \(-3\)
c) The coefficient of the term \(-x\) which can be written as \((-1)x\) is \(-1\)
d) The coefficient of the term \(y\) which can be written as \((1)y\) is \(1\)

Like Terms

Definition: Like terms are terms with the same variables raised to the same power. Constant terms are like terms
a) In the expression \(2x + 3y - 5x + 4 - 6y + 7\)
\(2x\) and \(-5x\) are like terms : same variable \(x\) to the same power 1
\(3y\) and \(-6y\) are like terms: same variable \(y\) to the same power 1
\(4\) and \(7\) are constants and therefore like terms
b) In the expression \(2xy - 3yx - 5x^{2} + 4 - 6y^{2} + 7\)
\(2xy\) and \(-3yx\) are the only like terms
c) In the expression \(9x^{2}y + 6yx + 5x^{2}y + 4 + 4y^{2}\)
\(9x^{2}y\) and \(5x^{2}y\) are the only like terms

Answer the Following Questions

  1. List all terms in and give the coefficient and degree for each in the algebraic expression below.
    1. \(2x + 2\)
    2. \(x + 5y - 9\)
    3. \(-x - y - 7\)
    4. \(2x^{2} - 9\)
    5. \(-xy^{2} - 9x + 6\)
    6. \(x^{2}y^{2} - 6x^{3} + 8\)

  2. Equivalent expressions have the same terms. Which of the following expressions are equivalent?
    1. \(2x + 2y - 3\)
    2. \(2x^{2} - 9\)
    3. \(2y^{2} + 2x - 3\)
    4. \(-9 + 2x^{2}\)
    5. \(2y - 3 + 2x\)
    6. \(3 + 2y^{2} + 8x\)
    7. \(8x - 3 - 2y^{2}\)
    8. \(-3 - 2y^{2} + 8x\)
    9. \(-3 + 2y + 2x\)

  3. List all like terms in each of the following expressions?
    1. \(2x - 2y + 10\)
    2. \(8x - 5y + 7 - 2x\)
    3. \(2x + 2y + 7 + 3x + 6y - 8\)
    4. \(x + 7 - x + 4\)
    5. \(2x^{2} + 5 - x^{2} - 3\)
    6. \(2y^{2}x - yx + 6 + 5xy^{2} + 3xy - 3\)

Solutions to the Above Questions

  1. Solution
    Terms are separated by operators. The coefficient and degree of a term has been defined above.

        expression       terms in the expression terms with
    coefficients and
    exponents highlighted     		    coefficient of each term    
    (red)    		 degree of
    each term    
    (blue)
    \(2x + 2\) \(2x\)
    \(+2\)     		\((2)x^{1}\)
    \(+2\)     		2
    constant term    		 1
    0
    \(x + 5y - 9\) \(x\)
    \(+5y\)
    \(-9\)     		\((1)x^{1}\)
    \((+5)y^{1}\)
    \(-9\)    		 1
    +5
    constant term     		1
    1
    0
    \(-x - y - 7\) \(-x\)
    \(-y\)
    \(-7\)    		 \((-1)x^{1}\)
    \((-1)y^{1}\)
    \(-7\)    		 -1
    -1
    constant term     		1
    1
    0
    \(2x^{2} - 9\) \(2x^{2}\)
    \(-9\)     		\((2)x^{2}\)
    \(-9\)    		 2
    constant term     		2
    0
    \(-xy^{2} - 9x + 6\) \(-xy^{2}\)
    \(-9x\)
    \(+6\)    		 \((-1)x^{1}y^{2}\)
    \((-9)x^{1}\)
    \(+6\)    		 -1
    -9
    constant term     		1 + 2 = 3
    1
    0
    \(x^{2}y^{2} - 6x^{3} + 8\) \(x^{2}y^{2}\)
    \(-6x^{3}\)
    \(+8\)     		\((1)x^{2}y^{2}\)
    \((-6)x^{3}\)
    \(+8\)    		 1
    -6
    constant term     		2+2 = 4
    3
    0

  2. Solution
    To compare algebraic expressions, we first order them from the term with the highest degree to the lowest.

    expression ordered
    a) \(2x + 2y - 3\) = \(2x + 2y - 3\)
    b) \(2x^{2} - 9\) = \(2x^{2} - 9\)
    c) \(2y^{2} + 2x - 3\) = \(2y^{2} + 2x - 3\)
    d) \(-9 + 2x^{2}\) = \(2x^{2} - 9\)
    e) \(2y - 3 + 2x\) = \(2x + 2y - 3\)
    f) \(3 + 2y^{2} + 8x\) = \(2y^{2} + 8x + 3\)
    g) \(8x - 3 - 2y^{2}\) = \(-2y^{2} + 8x - 3\)
    h) \(-3 - 2y^{2} + 8x\) = \(-2y^{2} + 8x - 3\)
    i) \(-3 + 2y + 2x\) = \(2x + 2y - 3\)

    We now use the ordered expressions on the right to compare the given expressions on the left of the table.

    Expressions a), e) and i) are equivalent

    Expressions b) and d) are equivalent

    Expressions g) and h) are equivalent

    Expressions c) and f) have no equivalent

  3. Solution
    Like terms have same variable with the same exponents. Numbers are like terms.

    expression like terms in groups
    a) \(2x - 2y + 10\) no like terms in this expression
    b) \(8x - 5y + 7 - 2x\) \(8x\) , \(-2x\) are like terms
    c) \(2x + 2y + 7 + 3x + 6y - 8\) \(2x\) and \(3x\) are like terms
    \(2y\) and \(6y\) are like terms
    \(7\) and \(-8\) are like terms
    d) \(x + 7 - x + 4\) \(x\) and \(-x\) are like terms
    e) \(2x^{2} + 5 - x^{2} - 3\) \(2x^{2}\) and \(-x^{2}\) are like terms
    \(+5\) and \(-3\) are like terms
    f) \(2y^{2}x - yx + 6 + 5xy^{2} + 3xy - 3\) \(2y^{2}x\) and \(+5xy^{2}\)
    \(-xy\) and \(+3xy\) are like terms
    \(6\) and \(-3\) are like terms

References and Links