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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

Multiplying powers Add the exponents.

(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

Adding powers with the The powers are like terms. Add the

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.

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

Multiplying powers Add the exponents.

(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

Adding powers with the The powers are like terms. Add the

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

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

= 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

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

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

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