Algebraic expressions

Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises

Algebraic expressions

Matematicas 2o E.S.O.Alberto Pardo Milanes

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1 Monomials

2 Operations with monomials

3 Polinomials

4 Operations with polynomials

5 Multiplying polynomials

6 Exercises

Alberto Pardo Milanes Algebraic expressions


Monomials



Monomials

Numerical value of an algebraic expression

An algebraic expression in variables x, y, z, a, r, t . . . k is anexpression constructed with the variables and numbers usingaddition, multiplication, and powers.To evaluate the numerical value of an algebraic expression meansthat you have to replace the variable in the expression with valuesand simplify the expression.

Example: To find the value of the algebraic expression x2−3x+4if x = 3, you replace every x by 3 and simplify:32 − 3 · 3 + 4 = 9− 9 + 4 = 4.



Monomials

A number multiplied with a variable in an algebraic expression isnamed coefficient. The product of a coefficient and one or morevariables is called a monomial.

Examples: x, 3xy2 and2

5x2y3z are all monomials, the coefficients

are 1, 3 and2

5.

In a monomial with only one variable, the power is called its order,or sometimes its degree. In a monomial with several variables, theorder/degree is the sum of the powers.

Examples: Deg(2x4)=4, Deg(7x3y2)=5.

Like monomials are monomials that have the exact same variables,but different coefficients. Unlike monomials are monomials that arenot like monomials.

Examples: 2x3y2 and2

5x3y2 are like monomials. 4xy2 and 4y2x4

are unlike monomials.



Operations with monomials




(Adding and subtracting monomials)

You can ONLY add and subtract like monomials. To add orsubtract like monomials, add or subtract the coefficients and keepthe variables.

Examples: 3x+4x = (3+4)x = 7x, and 20a−24a = (20−24)a == −4a.

(Multiplying monomials)

To multiply monomials, multiply the coefficients and add theexponents with the same bases.

Examples: 3x2 · 5y = (3 · 5)x2 · y = 15x2y, and 2a2 · 7ab4 == (2 · 7)a2 · ab4 = 14a2+1b4 = 14a3b4.




(Dividing monomials)

To divide monomials, divide the coefficients and subtract theexponents with the same bases.

Example: 15x3y3z2 : 3xy3z = (15 : 3)x3−1y3−3z2−1 = 5x2z.



Polinomials



Polinomials

A polynomial is a mathematical expression involving a sum ofpowers in one or more variables multiplied by coefficients. Apolynomial in one variable is given by a sum of several monomials.

Example: 3x2 − 5x− 2.

In a polynomial with only one variable, the highest power is calledits order, or sometimes its degree.

Example: Deg(x2 + 3x4 − 2x3 + 1) = Deg(3x4) = 4.

In a polynomial with several variables, the order/degree is thehighest sum of the powers of every term.

Example: Deg(xz3 + 3yx2z2 − 2) = Deg(yx2z2) = 5.



Operations with polynomials



Operations with polynomials

Add and subtract

(Adding polynomials)

Adding polynomials is just a matter of combining like monomials.Remenber you can only add like monomials.

Example: (7x2−x+4)+(x2−2x+3) = 7x2−x+4+x2−2x+3 == 9x2 − 3x+ 7.

(Subtracting polynomials)

To subtract a polynomial use the opposite of every coefficient ofthe subtrahend and add like monomials.

Example: (x3 + 3x2 + 5x− 4)− (3x3 − 8x2 − 5x+ 6) == x3 + 3x2 + 5x− 4− 3x3 + 8x2 + 5x− 6 == −2x3 + 11x2 + 10x− 10



Multiplying polynomials




(Multiply polynomials)

To multiply two polynomials, we multiply each monomial of onepolynomial (with its sign) by each monomial (with its sign) of theother polynomial. Write these products one after the other (withtheir signs) and then add like monomials to form the completeproduct.

Example: (x+ 3)(−2x+ 2) = (x+ 3)(−2x) + (x+ 3)2 == (−2x2 − 6x) + (2x+ 6) = −2x2 − 4x+ 6.




Multiply it vertically

Sometimes doing it vertically can be nicer:Example:(4x2 − 4x− 7)(x+ 3) = (4x2 − 4x− 7)x+ (4x2 − 4x− 7)3 =4x3 − 4x2 − 7x + 12x2 − 12x− 21 = 4x3 + 8x2 − 19x− 21

4x2 − 4x − 7× x + 3

12x2 − 12x − 214x3 − 4x2 − 7x

4x3 + 8x2 − 19x − 21



Exercises



Exercises

Exercise 1

Convert the statements into an algebraic expression using avariable and a sum or a difference:

• A number plus four:

• Five more than a number:

• A number minus five:

• The sum of a number and two:

• A number increased by ten:

• One less than a number:

• Seven added to a number:

• The difference of a number and eight:

• Nine less than a number

• A number decreased by three:

• Six subtracted from a number:

• The age a boy was two years ago:Alberto Pardo Milanes Algebraic expressions


Exercises

Exercise 2

Convert the statements into an algebraic expression using avariable and a multiplication or a division:

• Double a number:

• The quotient of a number and six:

• The product of four and a number:

• Twice a number:

• Nine divided by a number:

• A number multiplied by negative five:

• One fifth of a number:

• Three times a number:

• The ratio of a number to four:

• Eighty percent of a number :



Exercises

Exercise 3

Write the sentence as an algebraic expression:

• My bedroom’s lenght is 2 more feet than its width n. Thelenght is . . .

• The temperature at noon was t and had risen 8 degrees sinceseven o’clock. The temperature at 7:00 was . . .

• Lou charges 6,50 euros an hour to baby-sit. Today he workedx hours which means that he earned . . .

• I have y stamps from Asia and I have seven fewer stampsfrom Europe than from Asia. The total number of stamps Ihave is . . .



Exercises

Exercise 4

Write the sentence as an algebraic expression and operate:

• The base of a rectangle is double than the height. The area ofthe rectangle is. . .

• The product of a number and the number than comes after itis. . .

• I have nine fewer coins from China than from Australia. Thetotal number of coins I have is . . .

• Tom’s age is double than Fred’s age. The product of theirages is. . .

• The sum of a number and twice the number that comesbefore it is. . .



Exercises

Exercise 5

Find the degree of these monomials:Deg(5x4) =Deg(4y) =

Deg(1

2z3) =

Deg(abch2) =Deg(4xy) =Deg(x2) =Deg(33x7) =Deg(5a4b) =

Deg(3

5x2y) =

Deg(3x2y2) =



Exercises

Exercise 6

Link like monomials:7x 4x2y

2xy 7xy

3

5x2y

1

2x

3xy2 2xy2



Exercises

Exercise 7

Complete:Monomial Coefficient Order Variables

−7ab23

2m3n2p2

−7√3x3y4



Exercises

Exercise 8

Find:

2x2 · 3y = 2x2y3 · 74x2y = 4xa : 10a =

3

5x · 15y = 5xa · 10yb = 9

2x2y3 :

3

2xy =

1

2x2y · 3y = 4z · 5zy = 9zab2 : 6z =



Exercises

Exercise 9

Find the degree of these polynomials:Deg(x4 − 3x3 + 2x2 + 1) =

Deg(x2 − 3x3 + 2x+ 1) =

Deg(xy − x2 + yx3 + y2) =

Deg(x4y2 − 3x5) =



Exercises

Exercise 10

Find:(4x3 − 2x+ 5) + (x2 − 2x+ 1) + (−4x2 − 6) =

(7x3 + 2x2 + 4x+ 9)− (2x3 − 3x+ 8) =

(2x− 3) · (x+ 12) =

(x3 + 3x2 + 3x+ 1) · (x2 + 1) =


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Algebraic expressions

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