ALGEBRAIC EXPRESSIONS
Terminology
exponent
2x
Base (variable = letter)
Coefficient (number attached to the variable)
Term:Consists of constants and variables either multiplied or divided by each other.
E.g. 5xy,, and 3z are all single terms.
Terms are separated from each other by plus and minus signs.
LIKE TERMS are terms that have the same variables raised to the same powers,
e.g. 5a²bc and 3a²bc are like terms but 4a²b and 4ab' are unlike terms.
We can only add or subtract like terms, e.g.
5ax2 +3ax' =8ax? but
5ax? + 3ax cannot be simplified further
Algebraic expression: One term or acombination of terms separated by +and - signs.
Polynomial: An algebraic expression with one or more terms where the exponents of the
variables are natural numbers and there are no variables in the denominator of fractions. If
the exponents are not natural numbers, then the expression is not a polynomial.
The following ARE polynomials:
1. 2x2+ 3x + 5 is apolynomial because allthe exponents are natural numbers.
2.+3 is a polynomial because the exponent on the x is 1 which is a natural number.
3. 3x? is apolynomial because * =3x
The following are NOT polynomials:
1 x+3+ is NOT apolynomial because = 5x- and -1 is not anatural number.
2 2x2+ 3x +5 is NOT apolynomial because 2 is not anatural number.
3 V4x IS NOT a polynomial because v4x = 2x7 and-2 isnot a natural number.
Monomial: A polynomialwith one term, e.g. 4x
58 | Page
,Binomial: Apolynomial with 2terms, e.g. 6x-2
Trinomial: Apolynomial with 3terms, e.g. 5x2-3x- 1.
Coefficient: The number (including the sign) in front of the variable.
Constant term: Anumber (including the sign) on its own (without a variabie)
Degree of a polynomial: The highest exponent of a variable in a polynomial.
Examples:
1) 2a 4a - 6a + 12 is athird degree polynomial in a.
2) 3x*y²+ 4xy³ - 10x³y is afourth degre polynomial in x. We can also say it iSa
third degree polynomial in y.
Writing an expression in descending/ascending powers of x
Consider the expression: 3 -3x5 +x6 -2x2-8x4
(a) What is the degree of the polynomial?
(b) Write the expression in decreasing powers of x.
3c-8x23
x.
(c) Write the expression in ascending powers of
Example
-4x-3 + x +4
Consider the expression:
the polynomial. 3
a) Write down the degree of
decreasing powers of x.
b) Write the polynomial in
x-b'+-3 Why learn algebra?
coefficient of:
Finding Xisonly
c) What is the numerical useful if you're a
Yat
i) pirate!
i)
x3
iii)
constant term. - S
d) Write down the value of the
, subtracting terms
Adding and
only add or subtract like terms. the same
exponents.
Wecan variables raised to
Like terms have the same NOT add the exponents.
>
adding/subtracting terms, DO
> N.B. When
a
a'- 10ab + ab-
E.g.: Simplify 5a + 2a2 +
9ab
= 4a + 3a
3a??
Why do we not add 4a +
the following: 4a - 9ab?
Why do we not subtract
subtracting polynomials
Adding and
polynomials: 3a' 4a +5; -4a'-a+3
following
Example 1: Add the in brackets witha+ sign in betwe
(-4a-a+ 3) Place each polynomial
(3a2 4a + 5) + brackets,
4a-a+3 Simplify by removing the
5-
= 3a2- 4a +
Add / subtract like terms
=-a' 5a +8
+3
Subtract 3a'- 2a from 4a2- 6a
Example 2:
-(3a' - 2a) (distribute the negative sign)
= (4a' -6a + 3)
+3 3a2+ 2a
= 4a2 -6a
S FOR SECOND
= a'- 4a +3 AND FFOR
FIRST
Terminology
exponent
2x
Base (variable = letter)
Coefficient (number attached to the variable)
Term:Consists of constants and variables either multiplied or divided by each other.
E.g. 5xy,, and 3z are all single terms.
Terms are separated from each other by plus and minus signs.
LIKE TERMS are terms that have the same variables raised to the same powers,
e.g. 5a²bc and 3a²bc are like terms but 4a²b and 4ab' are unlike terms.
We can only add or subtract like terms, e.g.
5ax2 +3ax' =8ax? but
5ax? + 3ax cannot be simplified further
Algebraic expression: One term or acombination of terms separated by +and - signs.
Polynomial: An algebraic expression with one or more terms where the exponents of the
variables are natural numbers and there are no variables in the denominator of fractions. If
the exponents are not natural numbers, then the expression is not a polynomial.
The following ARE polynomials:
1. 2x2+ 3x + 5 is apolynomial because allthe exponents are natural numbers.
2.+3 is a polynomial because the exponent on the x is 1 which is a natural number.
3. 3x? is apolynomial because * =3x
The following are NOT polynomials:
1 x+3+ is NOT apolynomial because = 5x- and -1 is not anatural number.
2 2x2+ 3x +5 is NOT apolynomial because 2 is not anatural number.
3 V4x IS NOT a polynomial because v4x = 2x7 and-2 isnot a natural number.
Monomial: A polynomialwith one term, e.g. 4x
58 | Page
,Binomial: Apolynomial with 2terms, e.g. 6x-2
Trinomial: Apolynomial with 3terms, e.g. 5x2-3x- 1.
Coefficient: The number (including the sign) in front of the variable.
Constant term: Anumber (including the sign) on its own (without a variabie)
Degree of a polynomial: The highest exponent of a variable in a polynomial.
Examples:
1) 2a 4a - 6a + 12 is athird degree polynomial in a.
2) 3x*y²+ 4xy³ - 10x³y is afourth degre polynomial in x. We can also say it iSa
third degree polynomial in y.
Writing an expression in descending/ascending powers of x
Consider the expression: 3 -3x5 +x6 -2x2-8x4
(a) What is the degree of the polynomial?
(b) Write the expression in decreasing powers of x.
3c-8x23
x.
(c) Write the expression in ascending powers of
Example
-4x-3 + x +4
Consider the expression:
the polynomial. 3
a) Write down the degree of
decreasing powers of x.
b) Write the polynomial in
x-b'+-3 Why learn algebra?
coefficient of:
Finding Xisonly
c) What is the numerical useful if you're a
Yat
i) pirate!
i)
x3
iii)
constant term. - S
d) Write down the value of the
, subtracting terms
Adding and
only add or subtract like terms. the same
exponents.
Wecan variables raised to
Like terms have the same NOT add the exponents.
>
adding/subtracting terms, DO
> N.B. When
a
a'- 10ab + ab-
E.g.: Simplify 5a + 2a2 +
9ab
= 4a + 3a
3a??
Why do we not add 4a +
the following: 4a - 9ab?
Why do we not subtract
subtracting polynomials
Adding and
polynomials: 3a' 4a +5; -4a'-a+3
following
Example 1: Add the in brackets witha+ sign in betwe
(-4a-a+ 3) Place each polynomial
(3a2 4a + 5) + brackets,
4a-a+3 Simplify by removing the
5-
= 3a2- 4a +
Add / subtract like terms
=-a' 5a +8
+3
Subtract 3a'- 2a from 4a2- 6a
Example 2:
-(3a' - 2a) (distribute the negative sign)
= (4a' -6a + 3)
+3 3a2+ 2a
= 4a2 -6a
S FOR SECOND
= a'- 4a +3 AND FFOR
FIRST
