Download Algebraic Notation. Polynomials.

Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona
Algebraic notation and Polynomials
Algebraic Notation
The ability to convert worded sentences and problems into algebraic symbols and to
understand algebraic notation is essential in the problem solving process.
Notice that:
•
•
•
2 + 3
2 + 3 = 8
2 + 3 > 28
Is an algebraic expression, whereas
Is an equation, and
Is an inequality or inequation.
When we simplify repeated sums, we use product notation:
For example:
+
= 2 ‘lots’ of =2 × =2
and
++
= 3 ‘lots’ of =3 × =3 When we simplify repeated products, we use index notation:
For example:
× = and
× × = EXAMPLE 1
Write, in words, the meaning of:
a) − 5
b) + c) 3 + 7
a) Is “5 less than x”
b) Is “the sum of a and b” or “b more than a”
c) Is “7 more than three times the square of x”
EXAMPLE 2
Write the following as algebraic expressions:
a) The sum of p and the b) The square of the sum of p
square of q
a) + b) ( + )
c) 2 − and q
c) b less than double a
1
Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona
Algebraic notation and Polynomials
2
EXAMPLE 3
Write, in sentence form, the meaning of:
a) = b)
a) D is equal to the product of c and t
=
b) A is equal to a half of the sum of b and c,
or, A is the average of b and c.
EXAMPLE 4
Write “S is the sum of a and the product of g and t” as an equation.
The product of g and t is The sum of a and is + So, the equation is
= + TO PRACTICE
EXERCISE 1
Write in words, the meaning of:
a. 2
b. e. − 3
f.
b+c
g. 2 + h. (2)
j.
− k. + l.
i.
2
c. √
d. (a + b)
EXERCISE 2
Write the following as algebraic expressions:
a. The sum of " and #
i.
The difference between p and q if p>q
b. The sum of p, q and r
j.
a less than the square of b
c. The product of a and b
k. Half the sum of a and b
d. The sum of r and the square of s
l.
e. The square of the sum of r and s
m. A quarter of the sum of a and b
f.
n. The square root of the sum of m and n
The sum of the squares of r and s
g. The sum of twice a and b
h. The sum of x and its reciprocal
The sum of “a” and a quarter of “b”
o. The square root of the sum of the
squares of x and y
Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona
Algebraic notation and Polynomials
3
EXERCISE 3
Write in sentence form:
a. $ = + b. % =
d. ) = e. * = f.
+ = h. = √ + i.
=
-
g. % = , .
c. ' = 3(
&
&
EXERCISE 4
Write the following as algebraic equations:
a. S is the sum of p and r
b. D is the difference between a and b where b>a
The difference
between two
numbers is the
larger one minus
the smaller one
c. A is the average of k and m
d. M is the sum of a and its reciprocal
e. K is the sum of t and the square of s
f.
N is the product of g and h
g. Y is the sum of x and the square of d and e
Algebraic Substitution
To evaluate an algebraic expression, we substitute numerical values for the unknown, then
calculate the result.
Consider the number
crunching machine
alongside:
Input x
5x - 7
calculator
Output
If we place any number / into the machine, it calculates 0/ − 1. So, / is multiplied by 5, and
then 7 is subtracted:
For example: if = 2
5 − 7
=5×2−7
= 10−7
=3
and
if = −2 ;
5 − 7
= 5 × (−2) − 7
= −10 − 7
=−17
Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona
Algebraic notation and Polynomials
4
Notice that when we substitute a negative number such as −2, we place it in brackets. This
helps us to get the sign of each term correct.
TO PRACTICE
EXERCISE 5
If 4 = 3, = 1 and 5 = −2, evaluate:
7
b.
78
c.
7:98
f.
978
g.
a.
8
e.
(7:8)
9
7:9
9:7
8
9:87
78:9
d.
;8:9
h.
7
7:9
9
−4
EXERCISE 6
If = −3, = −4 and = −1, evaluate:
a. b. e. + f.
( + )
c. + d. ( + )
g. 2
h. (2)
EXERCISE 7
If = 4, = −1 and = = 2, evaluate:
a. > + b. > + e. >= − f.
> + c. >= − d. > − g. > + = + 2
h. >2 − 5=
Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona
Algebraic notation and Polynomials
DEFINITIONS: product, factors, sum, terms
A product is an expression where the last operation is multiplication.
In a product, the things being multiplied are called the factors.
A sum is an expression where the last operation is addition.
In a sum, the things being added are called the terms.
As an example, consider
the expression
then here is the order that computations would be done:
1. Add b and c .
2. Multiply this sum by a .
( + )
Notice that the last operation done is multiplication.
Thus, the expression ( + ) is a product.
The factors are and + .
As a second example,
consider the expression
here is the order that computations would be done:
1. Multiply a and b .
2. Add this result to c .
+ .
Notice that the last operation done is addition.
Thus, + is a sum.
The terms are and .
EXAMPLES
The expression 35 is a product.
The factors are @, /, A
The expression −4( + 2) is a product.
The factors are −B, /, / + C
The expression 5 − 5 + 1 is a sum.
The terms are0/, −A, D
The expression 2 + 253 − 7 is a sum.
The terms are /2, CA3, −1
EXERCISE 8
In the following expressions, how many terms are there? And each term has how many
factors?
a) 2 + 4 + 5( + )
b) E + 5FG + 2
5
Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona
Algebraic notation and Polynomials
6
Algebraic expressions
An algebraic expression is an expression that contains one or more numbers, one or more
variables, and one or more arithmetic operations.
It doesn't include an equal sign.
Algebraic expressions can be or many forms, for example:
3x 2 − 2 x − 1
3x 2 + 2 x +
1
x
− 6x 2 y − 2
3x 2 + 2 x
x2 − 3
A term consist of products of numbers and letters, so 3x2, -2x, -1, -6x2y, etc. are terms
The number multiplying the letters is the coefficient of the term.
3x2 (x2 term. Coefficient is 3)
-2x (x term. Coefficient is -2)
-1 (constant term is -1)
Polynomials
Polynomials are algebraic expresions. A polynomial in is a sum of terms, each of the form
k
,
where:
is a real number,
H is a nonnegative integer. That is, H = {0, 1,2, 3, . . . }.
DEFINITION: standard form of polynomials; degree;
leading coefficient
The standard form of a polynomial is:
"n/n + "n-1/n-1+. . . +"1/ + "0
Here, n denotes the highest power to which is raised; this highest power is
called the degree of the polynomial.
Thus, in standard form, the highest power term is listed first, and subsequent
powers are listed in decreasing order.
Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona
Algebraic notation and Polynomials
Notice that in the notation i i (read as " "KLMNONPQK/OROSQN "), the
number i denotes the coefficient of the i term.
The number n , which is the coefficient of the highest power term, is called
the leading coefficient of the polynomial.
Note that a constant (like 5 ) can be written as 5x0 .
This is why the power is allowed to equal zero in the definition of polynomial—to allow for
constant terms.
EXAMPLE 5
The expression 5x4-x3-3x2+7x-5
coefficient and degree.
is a polynomial. Find its terms, coefficients, leading
The terms are: 5x4, -x3, -3x2, 7x, and -5 .
Comparing each term with the required form ax k , we have:
term
writing in the
form ax k
a
k
5x4
5x4
a=5
k=4
-x3
(-1)x3
a=-1
k=3
-3x2
-3x2
a=-3
k=2
7x
7x
a=7
k=1
-5
-5x0
a=-5
k=0
Notice that every value of a is a real number, and every value of k is a
nonnegative integer.
The standard form of this polynomial is: 54−3−3 2+7 − 5
Here, the highest power term is written first, and subsequent terms decrease in
power.
The degree is 4 , since this is the highest power.
The leading coefficient is 5 , since this is the coefficient of the highest power term.
Notice that the leading coefficient actually leads (comes at the beginning of) the
polynomial, WHEN the polynomial is written in standard form.
EXAMPLE 6
The following expressions are NOT polynomials. Why?
a)
1x + x- 1
b) x - x1/2
c) 7x2 - 7x + 7x1/ 2
7
Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona
Algebraic notation and Polynomials
8
a) 1x + x- 1 is not a polynomial; no negative powers are allowed.
b) x - x1/2 is not a polynomial; the number 1/2 is not an allowable power.
c) 7x2 - 7x + 7x1/ 2 is not a polynomial; the number 1/2 is not an allowable power.
DEFINITION: monomial, binomial, trinomial
A polynomial with exactly one term is called a monomial.
A polynomial with exactly two terms is called a binomial.
A polynomial with exactly three terms is called a trinomial.
DEFINITION: quadratic, cubic, quartic
A polynomial of degree
A polynomial of degree
A polynomial of degree
A polynomial of degree
•
•
•
•
1
2
3
4
is called linear
is called quadratic.
is called a cubic.
is called a quartic.
Polynomials have beautiful smooth graphs—no breaks and no kinks.
The higher the degree of a polynomial, the more it is allowed to "turn" (change
direction).
Indeed, it can be shown easily (using calculus) that a polynomial of degree n can have
at most n-1 turning points.
The graph below is the polynomial T() = 3 − .
Notice that this polynomial has degree 3 and has 2 turning points.
Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona
Algebraic notation and Polynomials
EXERCISE 9
1) What is the degree of these polynomials?
a) 2x16+80x8
b) 7769x-97x7-56x9+31x19
c) 21x4-12x11+6x2-1710x10+4171x14
d) 60x3+65x5-3425x15
e) 8+27x11+234x18
2) What name is given to a polynomial with exactly one term?
3) What is the leading coefficient of these polynomials?
a)
b)
c)
d)
85-54x20
100x19-57x20+10029x6+2x4
61x4-6x3-7726x12+1425x11+45x19
9643x3-45x16-97x19
4) A polynomial is a sum of terms, each of a particular form. What is this form?
5) Is −28 an allowable term in a polynomial?
6) Is 2954 an allowable term in a polynomial?
7) Is −7x70 an allowable term in a polynomial?
8) Is the term x−9 an allowable term in a polynomial?
9) What is a quartic function?
10) What name is given to a polynomial with exactly two terms?
11) What is a trinomial?
12) What name is given to a polynomial with exactly one term?
13) Suppose that a polynomial has degree 8 . What (if anything) can be said about the
number of turning points for this polynomial?
14) Suppose that a polynomial has 5 turning points. What (if anything) can be said
about the degree of this polynomial?
15) Write the following polynomial in standard form:
a)
b)
c)
d)
e)
-3x14+8x27+6x5+4x6
4x2-7x4+5x6
-6x26+3x9-7x6
5x-2x25-2x2
2x3+8x6+8x7+4x-3x2
16) Can the graph of a polynomial have a break in it?
17) Can the graph of a polynomial have a kink in it?
9