UNIT 5 VOCABULARY: POLYNOMIALS

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UNIT 5 VOCABULARY: POLYNOMIALS

1.1. Algebraic Language

Algebra is a part of mathematics in which symbols, usually letters of the alphabet, represent numbers. Letters are used to represent quantities, so that we can express relationships between them by using the arithmetical, operations such as +, −, ×, ÷ and exponents. What do you do when you want to refer to a number that you do not

know? Suppose you want to refer to the number of buildings in your town, but you haven't counted them yet. You could say 'blank' number of buildings, or perhaps '?' number of buildings. In

mathematics, letters are often used to represent numbers that we do not know - so you could say 'x' number of buildings, or 'q' number of buildings.

These are called variables. Look at these examples:

•

The triple of a number: 3n

•

The triple of a number minus five units: 3n – 5

•

The following number to x: x + 1

•

The preceding number to y: y – 1

•

An even number: 2a

•

An odd number: 2z + 1 or 2z – 1 1.2. Algebraic expressions

An algebraic expression is a combination of numbers, variables, brackets, connected with operations.

Exercises. Find the algebraic expression for this sentences:

Sentence Algebraic Expression

I start with x, double it and then subtract 6. I start with x, add 4 and then square the result. I start with x, take away 5, double the result and

then divide by 3.

I start with x, multiply by 4 and then subtract t. I start with x, add y and then double the result.

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I start with n, square it and then subtract n. I start with x, add 2 and then square the result. A brick weighs x kg. How much do 6 bricks weigh?

How much do n bricks weigh?

A man shares x euros between n children. How much does each child receive?

1.3. Monomials

A monomial is an algebraic expression consisting of only one term.

A monomial can be any of the following:

• A constant: 2 4 -5 2

3 are monomials. • A variable: x y z

• The product of a constant and one or more variables: 2a 3xy 5x³ -x²y² THE VARIABLES OF A MONOMIAL SHOULD NOT HAVE NEGATIVE OR

FRACTIONAL EXPONENTS!

Therefore, these algebraic expressions are not monomials: 3x−2 _3xy1/2 3

t2

The coefficient of a monomial is the number that multiplies the variable(s). For example:

• The coefficient of 3x is 3.

• The degree of −x2y2z is -1 (because 2 + 2 + 1 =5).

The literal part of a monomial is formed by the variables (letters) and its exponents. For example:

• The literal part of 3x is x.

• The literal part of −x2y2z is x²y²z.

The degree of a monomial is defined as the sum of all the exponents of the variables. For example:

• The degree of 3x is 1 (the exponent of x is 1). • The degree of −x2y2z is 5 (because 2 + 2 + 1 =5). • The degree of 6 is 0 (there is no variable). Exercises.

1. Name the variables, coefficient, literal part and degree of the following monomials:

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2. Complete the table:

Monomial Variables Coefficient Literal part Degree

−3x2y3 7x3_yz −4 5x 3 32_x3 5 1.4. Polynomials

A polynomial is an expression which is made up of monomials that are added or subtracted.

Polynomials can have:

•

No variable at all (for example, 21 is a polynomial that has just one term, which is a constant).

•

One variable (for example x4 – 2x2 + x has three terms, but only one variable, x).

•

Two or more variables (for example, xy4 - 5x2z has two terms, and three variables ,x, y and z).

•

The degree of a polynomial with only one variable is the largest exponent of that variable.

•

A term that doesn't have a variable in it is called constant term.

•

The coefficient of the term with the highest degree is called the leading coefficient.

Exercises.

1. Complete this table:

Polynomial Degree Leading coefficient Constant term

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1 − x4 x2 2−x −x3_{−x− 7} 4 1.5. Evaluating a polynomial

To evaluate a polynomial, you plug in (substitute) the given value of x, and calculate the value.

For example, to evaluate P(x) = 2x3 – x2 – 4x + 2 at x = –3:

P(–3) = 2∙(–3)3 – (–3)2 – 4∙(–3) + 2 = 2∙(–27) – (9) + 12 + 2 = –54 – 9 + 14 = –63 + 14 = –49

ALWAYS REMEMBER TO BE CAREFUL WITH BRACKETS AND THE MINUS SIGNS!

Exercise. Evaluate x4 + 3x3 – x2 + 6

a) for x = –3 b) for x = –3 c) for x = 0 d) for x=1 2 2.1. Addition and subtraction of monomials

You can add or subtract monomials only if they have the same literal part (they are also called like terms). In this case, you sum or subtract the

coefficients and leave the same literal part. You can find some examples of like terms in the picture:

Look at these examples:

4xy² + 3xy² = 7xy² (we can add these monomials because they have the same literal part). -3xz + 7xz (we can't add these monomials because they don't have the same literal part).

2x² + 3 – 5x² + 1 = -3x² + 4 (and we can't add -3x² and 4 because they don't have the same literal part).

Exercise. Collect like terms to simplify each expression:

a) x² + 3x – x + 3x² c) (2x + 3)− (5x − 7)− (x −1) e) (x² + 2x) − (2x² − x) + (3x² + 5x) b) 5y + 3x + 2y + 4x d) 2 3x 2₊1 2x−32x 2₋1 5x+2 f) (x² + y) + (7x² − 3y) − (x² + 7y)

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2.2. Product of monomials

If you want to multiply two or more monomials, you just have to multiply the coefficients, and add the exponents of the equal letters:

More examples:

2xy²· (-5x²y) = -10 x³ y³ 3a² · 2ab = 6a³ b

Exercise. Multiply:

a) 5x · 3x c) -2x · x³ e) -7x²y · xy² g) 2

3x

3_y⋅3x

i) x² · (-2x) · 3x³ b) 2x · 3x² d) 4xy · 2x²y f) 5x³y² · xy h) 3x · 2x² · 5x³ j) 3ab · (-2a²) 2.3. Quotient of monomials

If you want to divide two monomials, you just have to divide the coefficients, and subtract the exponents of the equal letters.

THE QUOTIENT OF MONOMIALS GIVES AN EXPRESSION THAT IS NOT ALWAYS A MONOMIAL! More examples:

10x³ : 2x = 5x² 8x

2

y :6 y

3

=

8x

2

_y

6 y

3

=

4x

2

3y

2 Exercise. Operate:

a) 15x³ : 3x² c) 6x : 2x³ e) 15a³b² : ab g) 5xy · 2x²y : 3x b) 2x4_{: 3x} _{d) 12a²b : 3a} _f) _3x5_y2_:3x2_y2 _{h) xy³ · 3x²y : 2x²y²}

3.1. Addition of polynomials

To add two polynomials, you have to follow these steps:

•

Place like terms together. • Add the like terms

For example, to add p(x) = 2x2 + 6x + 5 and q(x) = 3x2 - 2x - 1

•

Start with: 2x2 + 6x + 5 + 3x2 - 2x - 1 =

•

Place like terms together: (2x2 + 3x2) + (6x - 2x) + (5 - 1) =

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You could also add two polynomials in columns like this:

3.2. Subtraction of polynomials

To subtract polynomials, follow these steps:

•

First reverse the sign of each term you are subtracting (in other words, turn "+" into "-", and "-" into "+").

•

Add as usual.

For example, to subtract p(x) = 2x2 + 6x + 5 and q(x) = 3x2 - 2x - 1

•

Start with: 2x2 + 6x + 5 - (3x2 - 2x - 1) =

•

Reverse the signs of each term: 2x2 + 6x + 5 - 3x2 + 2x + 1 =

•

Add the like terms: = 2x2 – 3x2 + 6x + 2x + 5 + 1 = -x2 + 8x + 6 And again, you can also do it in columns:

Exercise. Given the polynomials: P(x) = 4x2 − 1 Q(x) = x3 − 3x2 + 6x − 2 R(x) = 6x2 + x + 1 S(x) = ₂1x2+4 _{T(x) =} 3 2x 2₊₅ U(x) = x2 + 2 Calculate: a) P(x) + Q (x) b) P(x) − U(x) c) P(x) + R (x) d) 2P(x) − R(x) e) S(x) + T(x) + U(x) f) S(x) − T(x) + U(x) 3.3. Multiplication of polynomials

Before multiplying polynomials, let us look at a simpler case first:

• Monomial times binomial

Multiply the monomial by each of the two terms, like this:

• Polynomial times polynomial

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▪

Multiply each term in the first polynomial by each term in the second polynomial.

▪

Always remember to add like terms. For example, to multiply (x + 2y) ∙ (3x – 4y + 5)

• We multiply each term in the first polynomial by each term in the second polynomial: 3x2 – 4xy + 5x + 6xy – 8y2 + 10y

• We add like terms:

3x2 + 2xy + 5x – 8y2 + 10y And once again, you can also do it in columns:

Exercise. Multiply: a) (x4 − 2x2 + 2) · (x2 − 2x + 3) b) (3x2 − 5x) · (2x3 + 4x2 − x + 2) 4.1. Special products SQUARE OF A SUM! Examples:

(

x +5)

2

=

x

2

+

2⋅

x⋅5+5

2

=

x

2

+

10x+25

(

2x+3)

2

=(

2x)

2

+

2⋅2x⋅3+3

2

=

4x

2

+

12x+9

SQUARE OF A DIFFERENCE!

₍

a−b)

2

₌

a

2

₋

2ab+b

2

Examples:

(

x

2

−

2)

2

=(

x

2

)

2

−

2⋅

x

2

⋅

2+2

2

=

x

4

−

4x

2

+

4 (

x −√3)

2

=

x

2

−

2⋅

x⋅√3+(√3)

2

=

x

2

−

2√

3⋅x +3

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SUM TIMES DIFFERENCE!

₍

a+b)(a−b)=a

2

₋

b

2 Examples:

(

x + 3

2

)

⋅

(

x −3

2

)

=

x

2

−

(

3

2

)

2

=

x

2

−

9

4 (

ab

2

+

5)(ab

2

−

5)=(ab

2

)

2

−

5

2

=

a

2

b

4

−

25

Exercises.

1. Expand the following expressions:

a) (x + 2)2 c) (3x + 7)² e) (x – 3)² g) (a + 2b)²

b) (x + 5)2 d)

(

x

2−2

)

2

f) (x² + 2)² h) (2x - 2y)²

2. Expand the following expressions:

a) (3x − 2) · (3x + 2) c) (3 + x) · (3 - x) b) (x + 5) · (x − 5) d) (x −

√

2)·(x+

√

2) 3. Express as a binomial: a) x² + 12x + 36 = (x + __ )² f) 4x² – 9 = b) x² – 16 = (x + __ ) ∙ (x - __ ) g) 64x² - 160x + 100 = c) 4x² – 20x + 25 = ( __ - 5 )² h) 9x² – 25 = d) x² + 14x + 49 = i) 25x² + 40x + 16 = e) x² – 1 = j) x2−x + 1 4 4.2. Difference between equation,expression, identity and formulae

In Algebra, you use letter symbols to represent unknowns in a variety of situations: EQUATIONS In an equation the letters stand for one or more

particular numbers (the solutions of the equation). For example:

2x + 1 = x – 2

EXPRESSIONS In an expression there is no equals sign. For example:

3x² + 2x – 1

IDENTITIES In an identity there is an equals sign, but the equality holds for all values of the unknown. For example:

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FORMULAE In a formula, letters stand for defined quantities or variables.

For example:

d = s · t. (d is distance, s is speed and t is time) Exercise. Separate the equations, the formulae, the identities and the expressions:

a) x ( x + 1 ) = x² + x c) V = I · R e) 7x + 11 = x – 9 g) A = π · r² b) 7y + 10 d) x² - 3x + 10 f) (x + 1)² = x² + 2x + 1 h) x² - 7x = 0