Definitions
Let's learn a few definitions.
Factor of the number
When two or more numbers are multiplied, each of the numbers is called a factor of the number.
For example, in the product 5 ∗ 11 = 55, 5 and 11 are factors of 55.
Coefficient
Each factor is the coefficient of the product of other factors.
For example, in a term 3xy
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3 is a coefficient of xy
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x is a coefficient of 3y
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y is a coefficient of 3x
-
xy is a coefficient of 3
Generally, the numerical part of a term is called the numerical term of its coefficient.
Thus in the term 3xy, 3 is the numerical coefficient.
Exponent
Sometimes, the products are written as powers.
For example, 4 ∗ 4∗ 4 is written as 43
4 ∗ 4 is written as 42
a ∗ a ∗ a is written as a3
In a3, 3 is called the exponent or power and 'a' is called the base; the exponent 3 dictates the number of times the
base 'a' occurs as a factor in the product.
Monomial
A monomial is a term that is either a number or a variable with positive integral index or an indicated product of a
number and one or more variables.
Examples:
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7 is a monomial since it is a number.
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p is a monomial since it is a variable.
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7p is a monomial since it is an indicated product of a number 7 and p.
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7pq is also a monomial since it is a product of 7 and the variable 'pq'. 3/4 x2 y3 is also a monomial.
Polynomial
A polynomial is an indicated sum of monomials.
Examples:
Degree of polynomials
The degree of a polynomial is the greatest degree of its various terms.
Example:
2x + 3 has two terms, namely 2x and 3. The degree of 2x is 1. The degree of 3 is 0. The greatest of the two
degrees is 1.
Addition properties of polynomials
- A and B are two polynomials. By adding them, we get (A+B), which is also a polynomial. Hence, the set of polynomials has the Closure Property.
- A + B = B + A (Commutative Property)
- (A + B) + C = A + (B + C) (Associative Property)
- The zero polynomial is the identity element under addition.
- If 'A' is a polynomial, its additive inverse is -A. Thus, every polynomial has an additive inverse.
Example:
If A = 3x3 + 4x2 - x - 1
B = 4x3 -3 x2+ 4x + 5
Find A + B and B + A
A + B = (3x3 + 4x2 - x - 1) + (4 x3 - 3 x2+ 4x + 5)
= (3 + 4) x3 + (4 - 3) x2 + (- 1 + 4) x + (- 1 + 5)
= 7x3 + x2+ 3x + 4
B + A = ( 4x3 - 3x2 + 4x + 5) + (3x3 + 4x2 - x - 1)
= (4x3 + 3x3) + (- 3x2 + 4x2) + (4x - x) + 5 - 1
= 7x3 + x2 + 3x + 4
It can be seen that A + B = B + A.
Try these questions
State the coefficients and degrees of the polynomials.
-
10x5
Answer: Coefficient = 10; degree = 5
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- 2.51 x4
Answer: Coefficient = -2.51; degree = 4
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– 8
Answer: Coefficient = - 8; degree = 0
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√3x2
Answer: Coefficient = √3; degree = 2
Find the value of monomial when x = 3, 4.
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2x2
Answer: when x = 3
2 ∗(3)2 = 2 ∗9 = 18
when x = 4
2 ∗(4)2 = 2 ∗16 = 32
Find the values of the monomials when x = 2, 3, - 1.5
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3x2
Answer: When x = 2,
the value of
3x2 = 3 ∗(2)2 = 12
When x = 3;
the value of 3x2
= 3 ∗(3)2= 27
When x = - 1.5,
the value of 3x2
= 3 ∗(-1.5)2 = 6.75
-
-1.2 x2
Answer: When x = 2,
the value of -1.2 x2
= -1.2 ∗(2)2 = -4.8
When x = 3,
the value of -1.2 x2
= -1.2 ∗(3)2= - 10.8
When x = - 1. 5
the value of -1. 2x2
= - 1.2 ∗( - 1.5 )2 = - 2.7
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1/2x3
Answer: When x = 2,
the value of 1/2 x3
= 1/2 ∗2 ∗2∗2 = 4
When x = 3 ;
the value of 1/2 x3
= 1/2 ∗3∗ 3 ∗3 = 13.5
When x = -1.5 the value of 1/2 x3
= 1/2 ∗-1.5 ∗-1.5 ∗-1.5
= -1.6875
-
2x3
Answer: When x = 2,
then the value of 2x3
= 2 ∗(2)3 = 2∗ 8 = 16
When x = 3,
then the value of 2x3
= 2 ∗(3)3 = 2 ∗27 = 54
When x = - 1.5,
then the value of 2x3
= 2 ∗(-1.5)3 = - 6.75
Simplify
-
- 3x2 + ( 6x2 ) - ( -0.5x2 ) + ( 1.5x2)
Answer: = - 3x2 + 6x2 + 0.5x2 + 1.5x2
= ( - 3 + 6 + 0.5 + 1.5 ) x2 = 5x2
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( - 3x ) + ( - 4x ) - ( 4.5 ) x + ( 2.5x )
Answer: = ( - 3 - 4 - 4.5 + 2.5 ) x = - 9x
-
( 3x ) + ( - 4x ) - ( - 3x ) + ( - 7x )
Answer: = 3x - 4x + 3x - 7x
= (3 - 4 + 3 - 7) x = - 5x
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( - 5x2 ) + ( 5.2x2 ) + ( 1.5x2 ) - ( 0.7x2 )
Answer: = ( - 5 + 5.2 + 1.5 - 0.7 ) x2
= ( 6.7 - 5.7 ) x2 = (1) x2 = x2