# Operations with Algebraic Expressions

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An algebraic expression is one or more algebraic terms in a phrase. It can include variables, constants, and operating symbols, such as plus and minus signs. It's only a phrase, not the whole sentence, so it doesn't include an equal sign.
1. Operations with Algebraic Expressions:
Multiplication of Polynomials
The product of a monomial x monomial
To multiply a monomial times a monomial, multiply the coefficients and add the exponents
on powers with the same variable as a base.
Note: A common mistake that many students make is to multiply the exponents on powers with
the same variables as a base. This is NOT CORRECT. Remember the exponent rules!
Exponent Rules
Case What to do Rule Example
(xa)(xb) = xa+b (25)(23) = 28
with the same base Keep the base the
same.
Dividing powers with Subtract the xa = xa ÷ xb = xa−b 25 = 25 ÷ 23= 22
the same base exponents. Keep the
xb 23
base the same.
Simplifying power of a Multiply the exponents. (xa)b = xab (25)3 = 215
power Keep the base the
same.
Exponent of 0 Anything to the x0 = 1 20 = 1
exponent of 0 equals 1.
Other cases that come up when working with powers
Case What to do Example
xa + xa = 2xa
same base and SAME coefficients; keep the base and the
exponents exponent the same.
Adding powers with the The powers are NOT like terms. They
xa + xb = xa + xb
same base and DIFFERENT can NOT be added.
Subtracting powers with the The powers are like terms. Subtract
2xa − xa = xa
same base and SAME the coefficients; keep the base and the
exponents exponent the same.
Subtracting powers with the The powers are NOT like terms. They
xa − xb = xa − xb
same base and DIFFERENT can NOT be subtracted.
2. Multiplication of Polynomials
Example 1:
Simplify. 5x3 (−6x)
= 5(−6)x3 + 1 Multiply the coefficients. Add the exponents since both powers have base x.
= −30x4
Example 2:
1
Simplify. 20a2y−3b ( ay4)
4
1
= 20( )a2+1y−3+4b Multiply the coefficients. Add the exponents for powers with base a. Add
4
the exponents for powers with base y.
= 5a3yb
Example 3:
Simplify. 0.10p10q4 (10)p5q-4
= 0.10(10)p10+5q4-4 Multiply the coefficients. Add the exponents for powers with base
a. Add the exponents for powers with base y.
= 1p15q0 Simplify q0. Any number to the exponent of 0 equals 1.
= p15(1) p15 multiplied by 1 is just p15
= p15
The product of a monomial x binomial
To multiply a monomial by a binomial, multiply the monomial by EVERY term making up the
Remember: To find the product of two terms, multiply the coefficients and add the
exponents on powers with the same variable as a base.
Example 1:
Expand. 5x3(7x2 + 15xy)
5x3(7x2 + 15xy) Step 1: Multiply the monomial by EVERY term making up the binomial.
= 5x3(7x2) + 5x3(15xy) Step 2: To find the product of two terms, multiply the coefficients
and add the exponents on powers with the same variable as a
base.
Tutoring and Learning Centre, George Brown College 2014 www.georgebrown.ca/tlc
3. Multiplication of Polynomials
= 35x5 + 75x4y
Example 2:
Expand. −2c4p(10c3p2 – 4c)
−2c4p(10c3p2 – 4c) Step 1: Multiply the monomial by EVERY term making up
the binomial.
= −2c4p(10c3p2) + (−2c4p)( –4c) Step 2: To find the product of two terms, multiply the
coefficients and add the exponents on powers with the
same variable as a base.
= −20c7p3 + 8c5p
The product of a monomial x trinomial OR monomial x polynomial
To multiply a monomial by a trinomial or any polynomial, multiply EVERY term in the
trinomial or polynomial by the monomial.
To find the product of two terms, multiply the coefficients and add the exponents on powers with
the same variable as a base.
Expand. 7x3(19x7y + 20 – 3x + y)
7x3(19x7y + 20 – 3x + y) Step 1: Multiply the monomial by EVERY term making up
the binomial.
= 7x3(19x7y) + 7x3 (20) + 7x3 (−3x) + 7x3 (y) Step 2: To find the product of two terms,
multiply the coefficients and add the
exponents on powers with the same
variable as a base.
= 133x10y + 140x3 – 21x4 + 7x3y
The product of a binomial x binomial
To multiply a binomial by a binomial, multiply EVERY term in the first binomial by EVERY
term in the second binomial. Then simplify by collecting (adding or subtracting) like terms, if it
is possible.
Tutoring and Learning Centre, George Brown College 2014 www.georgebrown.ca/tlc
4. Multiplication of Polynomials
You can use the FOIL (First, Outer, Inner, Last) method to remember how to multiply binomials.
First Last
(a + b)(c + d) = ac + ad + bc + bd
Inner
Example 1: Outer
Expand. (3x + 2)(5x – 2)
(3x + 2)(5x – 2) Step 1: Use FOIL to multiply every term in the first
binomial by every term in the second binomial.
= 3x(5x) + (3x)(−2) + (2)(5x) + (2)(−2) Step 2: Evaluate every product.
= 15x2 + (−6x) + 10x + (−4)
= 15x2 −6x + 10x −4 Step 3: Collect like terms.
= 15x2 + 4x −4 This is the final answer.
Example 2:
Expand. (2y − 8)(3x – 1)
(2y − 8)(3x – 1) Step 1: Use FOIL to multiply every term in the first
binomial by every term in the second binomial.
= 2y(3x) + (2y)(−1) + (−8)(3x)+ (−8)(−1) Step 2: Evaluate every product.
= 6yx + (– 2y) + (–24x) + (8) There are no like terms that can be collected.
Simplify double signs. Arrange terms in
alphabetical order*.
= 6yx – 24x – 2y + 8 This is the final answer.
*Note: By convention, terms are written from highest to lowest degree and in alphabetical order..
Tutoring and Learning Centre, George Brown College 2014 www.georgebrown.ca/tlc
5. Multiplication of Polynomials
Squaring a binomial
To square a binomial means to multiply the binomial by itself.
The rules of multiplying a binomial by a binomial apply. To multiply a binomial by a binomial,
multiply EVERY term in the first binomial by EVERY term in the second binomial.
When multiplying a binomial by itself, the expanding follows a pattern as shown below.
(a + b)2 = (a + b)(a + b) = a2 + ab + ab+ b2 = a2 + 2ab + b2
Example 1:
Expand. (5x − 3)2
Solution 1:
(5x − 3)2 = (5x − 3)(5x − 3) Step 1: Use FOIL method to expand.
= 25x2 − 15x − 15x + 9 Step 2: Collect like terms.
= 25x − 30x + 9
2
Solution 2:
(5x −3)2 = (5x)2 + 2(5x)(−3) + (−3)2 Step 1: Use the (a + b)2 = a2 + 2ab + b2 pattern to expand.
(a + b)2 = a2 + 2ab + b2
= 25x2 – 30x + 9 Step 2: Simplify each term.
Example 2:
Expand. (9y + 2)2
(9y + 2)2 = (9y)2 + 2(9y)(2) + 22 Step 1: Use the (a + b)2 = a2 + 2ab + b2 pattern to expand
= 81y2 + 36y + 4 Step 2: Simplify each term.
Tutoring and Learning Centre, George Brown College 2014 www.georgebrown.ca/tlc
6. Multiplication of Polynomials
Practice Questions:
1. Simplify each of the following algebraic expressions:
a) 3b2a(–2b3)
b) –p4r(–21pr2)
c) 22a(b2a – 2b3a)
d) 19x(–y2 + 3x3z)
e) 3s(2s + 4qs2 –8)
f) –2(11x2 + 20xy –14y3)
g) –(28q3s9 + 2q2s –10q5s3 + 18q +9)
h) (2x + 3)(19x – 1)
i) (3x + 5)(7 – 3x)
j) (9y2 + 8)(3y –2)
k) (5y – 3)(y3 + 6)
l) (8a – 3)(9a – 10)
m) (9x + 2)2
n) (14 – y)2
o) (x2 – y)2
p) (4a3 – 3b)2
1. a) –6b5a l) 72a2 – 107a – 30
b) 21p5r3 m) 81x2 + 36x + 4
c) 22b2a2 – 44b3a2 n) y2 – 28y + 196
d) 57x4z – 19xy2 o) x4 – 2x2y + y2
e) 6s2 + 12qs3 – 24s p) 36a6 – 24a3b + 9b2
f) –22x2 – 40xy + 28y3
g) –28q3s9 – 2q2s + 10q5s3 – 18q – 9
h) 38x2 + 55x – 3
i) –9x2 + 6x + 35
j) 27y3 – 18y2 + 24y – 16
k) 5y4 – 3y3 + 30y – 18
Tutoring and Learning Centre, George Brown College 2014 www.georgebrown.ca/tlc