Grade 9: Algebraic Expressions
Updated 15 March 2013
Grade 9
Algebraic Expressions
Goals:
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Recognize and use conventions for writing algebraic expressions
Identify and classify like and unlike terms
Recognize and identify coefficients and exponents
Use the commutative, associative, and distributive laws
Use the laws of exponents
Multiply integers and monomials by:
o Monomials
o Binomials
o Trinomials
Divide the following by integers or monomials
o Monomials
o Binomials
o Trinomials
Find the product of two binomials
Find the square of a binomial
Simplify algebraic expressions
Determine squares, cubes, square roots and cube roots of single algebraic terms
Determine the numerical value of algebraic expressions by substitution
Terminology:
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Monomial
Binomial
Trinomial
Polynomial
Degree
Like/Unlike Terms
Coefficient
1
Grade 9: Algebraic Expressions
Polynomials
Polynomials
A polynomial is a mathematical expression involving the sum of powers of a
variable. The variable is often 𝑥, and each power of the variable can have a
coefficient. Here’s an example:
𝑥 3 + 7𝑥 2 − 3𝑥 + 1
In this case the variable is 𝑥 and the coefficients are 1, 7, −3, and 1. It is standard to
write polynomials in decreasing powers and with all coefficients written before the
power of 𝑥.
Naming
Polynomials are named according to their number of terms as follows:
Terms
1
2
3
4+
NUMBER OF TERMS
Example
Name
Monomial
Binomial
Trinomial
Polynomial
𝑥4
𝑥2 − 1
4
𝑥 + 𝑥3 + 𝑥 − 7
𝑥 5 − 3𝑥 3 + 2𝑥 2 + 𝑥 + 6
Exercise 1
1. Rewrite each polynomial so that the powers are in descending order. Then classify the
polynomial according to its number of terms (i.e. monomial, binomial, etc.)
1.1. 𝑥 3 + 3𝑥 + 2
1.2. 𝑝5 + 3𝑝 + 2𝑝2
1.3. 𝑦 7 − 𝑦 8 + 𝑦 9 − 𝑦10
1.4. 𝑥100
1.5. 𝑎2 − 1
1.6. 1 + 2𝑏 + 3𝑏 2 + 4𝑏 3 + 5𝑏 4
1.7. 7𝑥 7 + 16𝑥 8
1.8. 𝑎2 + 𝑎4 + 𝑎5
2. Consider the polynomial 4𝑥 3 + 19𝑥 − 3𝑥 4 :
2.1. Rewrite the polynomial in decreasing powers of 𝑥.
2.2. What is the coefficient in front of the 𝑥 3 term?
2.3. Classify the polynomial according to its number of terms.
2.4. What is the value of the expression when 𝑥 = 1?
2
Grade 9: Algebraic Expressions
3. Consider the polynomial 17𝑎3 − 𝑎5 + 𝑎4 :
3.1. Rewrite the polynomial in decreasing powers of 𝑎.
3.2. What is the coefficient in front of the 𝑎5 term?
3.3. Classify the polynomial according to its number of terms.
3.4. What is the value of the expression when 𝑎 = 0?
4. Consider the polynomial 𝑝4 − 1:
4.1. Rewrite the polynomial in decreasing powers of 𝑝.
4.2. What is the coefficient in front of the 𝑝4 term?
4.3. Classify the polynomial according to its number of terms.
4.4. What is the value of the expression when 𝑝 = 3?
5. Using the variable 𝑥, write a trinomial which has a value of 10 when 𝑥 = 2.
6. Using the variable 𝑝, write a binomial which has a value of 30 when 𝑝 = 3.
3
Grade 9: Algebraic Expressions
Simplifying Algebraic Expressions
Like and Unlike Terms – Within a
polynomial expression, like terms are
terms which have the same variable(s)
and exponent(s). For example, 3𝑥 2 and
−4𝑥 2 are like terms because they both
have 𝑥 2 . Their coefficients are 3 and −4
respectively.
Adding and Subtracting Polynomials.
When adding and subtracting
polynomials, the key is to add or
subtract the coefficients of like terms.
For example, consider the sum of 2𝑎
and 3𝑎. In this case, 2𝑎 and 3𝑎 are like
terms with coefficients 2 and 3, and 2 +
3 = 5, so 2𝑎 + 3𝑎 = 𝟓𝒂. More
examples are given at right:
Like Terms
3𝑎 and −4𝑎
𝑥 2 and 4𝑥 2
10𝑎𝑏 and 5𝑎𝑏
2𝑥 2 and −𝑥 2
NOT like terms
3𝑎 and −4𝑥
𝑥 2 and 4𝑥 3
10𝑎𝑏 and 5𝑎𝑏 2
2𝑥 2 and −𝑥 3
Terms
2𝑎 + 3𝑎
−7𝑥 3 + 4𝑥 3
9𝑥 + 10𝑥
𝑎𝑏 + 𝑎𝑏
𝑥−𝑥
𝑥3 + 𝑥2
𝑎𝑏 + 2𝑎𝑏 + 3𝑎𝑏
Simplification
5𝑎
−3𝑥 3
19𝑥
2𝑎𝑏
0
𝑥3 + 𝑥2
6𝑎𝑏
NB: Unlike terms CANNOT be simplified! For example, 𝑥 2 and 𝑥 3 CANNOT be combined
under addition and subtraction, but only under multiplication or division, which we
discussed in our chapter on exponents. That is, 𝑥 2 × 𝑥 3 = 𝑥 2+3 = 𝑥 5 , but 𝑥 2 + 𝑥 3 cannot
be simplified.
Exercise 2
1. Identify the like terms in the lists below:
1.1. 𝑎𝑏
𝑎𝑐
7𝑎𝑏
7𝑎
2
1.2. 𝑥
𝑥
−𝑥
𝑥3
1.3. 8𝑦
8𝑦 2
4
9𝑦
1.4. 𝑎
19
2
𝑎2
1.5. 2𝑎2
𝑧2
𝑧
2𝑧
2 2
2
2 2
1.6. 𝑎 𝑏
2𝑎 𝑏
2𝑎 𝑏
2𝑎𝑏 2
1.7. 9𝑥
9𝑦
9𝑧
9𝑥
1.8. 𝑎𝑏
𝑎𝑐
7𝑎𝑏
𝑏
2. Simplify each of the following like terms.
2.1. 2𝑎𝑏 + 2𝑎𝑏 + 2𝑎𝑏
2.2. 𝑥 + 𝑥 + 𝑥
2.3. 𝑥 2 + 𝑥 2 + 𝑥 2 + 𝑥 2
2.4. 𝑝 + 2𝑝 + 3𝑝 + 4𝑝
2.5.
2.6.
2.7.
2.8.
4
−3𝑦 + 5𝑦
4𝑎 − 7𝑎
9𝑧 − 6𝑧 − 3𝑧
9𝑥 − 4𝑥
Grade 9: Algebraic Expressions
3. Match each expression on the left with its correct simplification on the right:
Answer
Expression
Simplification
3.1
2𝑎𝑏 + 3𝑎𝑏
A
5𝑎2 𝑏
3.2
B
2𝑎2 𝑏 + 3𝑎2 𝑏
4𝑎2 𝑏 2
3.3
C
3𝑎𝑏 2 + 2𝑎𝑏 2
6𝑎2 𝑏 2
3.4
D
4𝑎4 𝑏 4
2𝑎2 𝑏 2 + 2𝑎2 𝑏 2
3.5
2𝑎𝑏 × 3𝑎𝑏
E
5𝑎𝑏
3.6
F
2𝑎2 𝑏 × 3𝑎2 𝑏
6𝑎4 𝑏 2
3.7
G
3𝑎𝑏 2 × 2𝑎𝑏 2
5𝑎𝑏 2
3.8
H
2𝑎2 𝑏 2 × 2𝑎2 𝑏 2
6𝑎2 𝑏 4
4. Simplify each of the following expressions. BE CAREFUL!
4.1. 𝑥 2 + 2𝑥 2 + 3𝑥 2
4.7. 8𝑦 + 9𝑦
4
3
4
3
4.8. 𝑥 + 𝑥 4
4.2. 𝑝 + 𝑝 − 𝑝 + 𝑝
4.9. 4𝑦 + 5𝑦 2 + 4𝑦 2
4.3. 𝑦 5 + 5𝑦 5
4.10. 𝑎 + 𝑐 + 2𝑎 + 𝑎𝑐
4.4. 𝑎 + 𝑏 + 𝑎𝑏 + 2𝑎 + 3𝑏 + 4𝑎𝑏
3
2
4.11. 5𝑎𝑏 − 𝑐 + 5𝑎𝑏 + 𝑐
4.5. 2𝑥 + 3𝑥
4.12. 9𝑎 + 10𝑎 + 3
4.6. 12𝑧 − 12𝑧 2 + 9𝑧
5. Add each of the following polynomials. Your final answer should be in order of
descending powers:
5.1. (𝑥100 + 2𝑥10 + 𝑥) + (−𝑥100 + 3𝑥10 − 3𝑥)
5.2. (𝑝2 − 𝑝 − 7) + (𝑝2 + 3𝑝 + 4)
5.3. (𝑎5 + 𝑎3 + 𝑎) + (−𝑎4 − 𝑎2 − 1)
5.4. (𝑧 3 − 𝑧 2 ) + (3𝑧 3 + 3𝑧)
5.5. (19𝑥 4 − 1) + (20𝑥 4 + 1)
5.6. (4𝑦 + 3) + (4𝑦 2 + 2)
5.7. (𝑥 3 + 𝑥 2 ) + (𝑥 2 + 𝑥 + 1)
5.8. (3𝑛3 + 2𝑛2 ) + (3𝑛3 + 2𝑛2 + 𝑛)
6. Match each expression on the left with its correct simplification on the right:
Answer
Expression
Simplification
6.1
𝑥+𝑥
A
𝑥2
6.2
𝑥×𝑥
B
𝑥3
6.3
2𝑥 + 2𝑥
C
6𝑥 2
6.4
2𝑥 × 2𝑥
D
4𝑥
6.5
3𝑥 + 3𝑥
E
6𝑥
6.6
3𝑥 × 2𝑥
F
𝑥 + 𝑥2
6.7
G
𝑥 + 𝑥2
4𝑥 2
6.8
H
2𝑥
𝑥 × 𝑥2
5
Grade 9: Algebraic Expressions
Multiplying Polynomials
Multiplying a Monomial and a Binomial
When multiplying a binomial and a monomial, we use the
distributive law and “distribute” the monomial to both terms
of the binomial. An illustration of the distributive law is on the
right.
Distributive Law: 𝑎 (𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐
𝑎𝑏
𝑏
𝑎𝑐
𝑐
Example: −2(𝑥 + 3) = (−2)(𝑥) + (−2)(3) = −2𝑥 − 6
𝑎
Multiplying Binomials
O
When multiplying binomials we use the distributive
law twice. That is, distribute the first term of the
𝑑 first binomial to the second binomial, then
distribute the second term of the first binomial to
the second binomial. An easier way to remember
𝑐 this is the acronym FOIL. We add the products of
the First terms, then the Outer terms, then the
Inner terms, then the Last terms
L
𝑎𝑑
F
𝑏𝑑
I
𝑎𝑐
𝑏𝑐
𝑎
𝑏
F
First
Outer
Inner
Last
L
I
O
F
L
Multiplying Binomials: (𝑎 + 𝑏)(𝑐 + 𝑑) = 𝑎𝑐 + 𝑎𝑑 + 𝑏𝑐 + 𝑏𝑑
I
O
Examples:
F
I
O
L
F
O
I
L
(𝑥 + 1)(𝑥 + 2) = (𝑥)(𝑥) + (𝑥)(2) + (1)(𝑥) + (1)(2) = 𝑥 2 + 2𝑥 + 𝑥 + 2 = 𝑥 2 + 3𝑥 + 2
F
O
I
L
(𝑝2 − 2)(𝑝 + 3) = (𝑝2 )(𝑝) + (𝑝2 )(3) + (−2)(𝑝) + (−2)(3) = 𝑝3 + 3𝑝2 − 2𝑝 − 6
6
Grade 9: Algebraic Expressions
Exercise 3
1. Distribute each monomial to its following polynomial
1.1. 𝑏(𝑏 + 1)
1.7.
1.8.
1.2. 𝑦(𝑦 − 1)
1.3. 2(𝑥 + 1)
1.9.
1.4. 3(2𝑐 + 3)
1.10.
2 2
1.5. 𝑥 (𝑥 + 𝑥 + 1)
1.11.
2
1.6. −2(𝑧 + 1)
1.12.
−(𝑥 + 1)
𝑎(𝑎4 + 𝑎3 + 𝑎2 + 𝑎 + 1)
−5(10𝑥 + 10)
3𝑝(𝑝2 + 𝑝 + 1)
−(𝑥 − 1)
3(2𝑥 2 + 4𝑥 + 2)
2. Use the included area diagrams to multiply the binomials
2.1. (1 + 𝑥)(𝑥 + 1) =
2.3. (2 + 5)(3 + 2) =
2.2.
1
𝑥
1
4
10
2
𝑥
𝑥2
𝑥
6
15
3
1
𝑥
2
5
(𝑎 + 3)(3 + 𝑎) =
2.4.
(1 + 2𝑐)(3𝑐 + 2)
𝑎2
3𝑎
𝑎
2
4𝑐
2
3𝑎
9
3
3𝑐
6𝑐 2
3𝑐
𝑎
3
1
2𝑐
3. Draw your own area diagram to multiply the following binomials:
3.1. (𝑥 + 2)(𝑥 + 3)
3.2. (2 + 𝑎)(2 + 𝑏)
3.3. (5 + 𝑥)(5 + 𝑥)
3.4. (𝑧 + 3)(𝑧 + 4)
4. Use FOIL to multiply the following binomials. Don’t forget to combine all like terms in
your final answer:
4.1. (𝑥 + 3)(𝑥 − 3)
4.6. (3 − 𝑦)2
4.2. (𝑧 + 1)(𝑧 + 2)
4.7. (𝑎 + 𝑏)2
4.3. (2𝑎 + 3)(𝑎 + 2)
4.8. (𝑥 + 8)2
4.4. (𝑎 + 𝑏)(𝑎 + 𝑏)
4.9. (9 − 𝑐)(9 + 𝑐)
2
4.5. (𝑥 + 2)
4.10. (2 + 𝑥)(𝑥 + 4)
N.B. For 4.5 → 4.8, remember that (𝑥 + 2)2 = (𝑥 + 2)(𝑥 + 2) and NOT 𝑥 2 + 4.
7
Grade 9: Algebraic Expressions
Algebraic Expression Problem Solving
Exercise 5
1. Which values of 𝑥 make the following expressions equal 0? Each question has the
number of real number solutions in brackets.
1.1. 𝑥 2 − 4
(2 real number answers)
3
1.2. 𝑝 − 27
(1 real answer)
2
1.3. 𝑥 − 𝑥 − 2
(2 real answers)
2
1.4. 𝑥 + 8𝑥 + 16 (1 real answers)
2. Multiply the following polynomials:
2.1. (𝑥 2 + 𝑥 + 1)(𝑥 + 1)
2.2. (𝑥 2 + 𝑥 + 1)2
2.3. (𝑥 2 + 𝑥 + 1)(𝑥 3 + 𝑥 2 + 𝑥 + 1)
3. Multiply the following polynomials:
3.1. (𝑥 + 1)(𝑥 − 1)
3.2. (𝑝2 − 1)(𝑝2 + 1)
3.3. (𝑎 + 𝑏)(𝑎 − 𝑏)
3.4. (2 + 𝑦)(2 − 𝑦)
3.5. What pattern emerges from 3.1 - 3.4?
4. Mr. Ebert drew 2 different area diagrams to multiply binomials, but he forgot what
binomials he was multiplying. Help him out by completing the diagrams
4.1.
4.2.
3
3𝑥
?
𝑥
𝑥2
?
?
?
60
10𝑏 2
?
6𝑎2
^2
𝑎2 𝑏 2
?
?
?
8
Grade 9: Algebraic Expressions
Appendix: Common Misconceptions
The following is a list of common mistakes student make while working with algebraic
expressions. If you make a mistake on your homework, it will most likely be one of these, so
watch out!
CORRECT STATEMENT
COMMON MISTAKE
−24 = −(2)4 = −16
and NOT +16.
(−2)4 = 16
and NOT −16
𝑥 + 𝑥 = 2𝑥
and NOT 𝑥 2
𝑥 3 + 𝑥 3 = 2𝑥 3
and NOT 𝑥 6
𝑎+𝑏 =𝑎+𝑏
and NOT 𝑎𝑏
(−2𝑥 2 )3 = −8𝑥 6
and NOT −6𝑥 5
−𝑥(3𝑥 + 1) = −3𝑥 2 − 𝑥
and NOT −3𝑥 2 + 𝑥
6𝑥 2 + 1
1
=6+ 2
2
𝑥
𝑥
and NOT 6 + 1
If 𝑥 = 2, then −3𝑥 2 =
and NOT (−6)2
−3(2)2 = −3(4) = −12
If 𝑥 = −2, then – 𝑥 2 − 𝑥 =
and NOT 4 + 2 = 6
−(−2)2 − (−2) = −4 + 2
√25𝑥 2 − 9𝑥 2 = √16𝑥 2 = 4𝑥
and NOT 5𝑥 − 3𝑥 = 2𝑥.
(𝑥 + 2)2 = 𝑥 2 + 4𝑥 + 4
and NOT 𝑥 2 + 4
9