Introduction about polynomial:
In mathematics, a polynomial is an expression of finite length constructed from variables (also known as indeterminate) and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. For example, x2 - 4x + 7 is a polynomial, but x2 - 4/x + 7x3/2 is not, because its second term involves division by the variable x and because its third term contains an exponent that is not a whole number. (Source: Wikipedia)
Example Problems for Determining Polynomial Functions :
Determining polynomial functions - Example problem 1:
Simplify f(x) = (21x + 15y) + ( 9x – 21y ) – (11x – 17y)
Solution:
f(x) = (21x + 15y) + ( 9x – 21y ) – (11x – 17y) = 21x + 15y + 9x – 21y – 11x + 17y
= 21x + 9x - 11x + 15y – 21y + 17y
= 19x +11y
The answer for f(x) = (21x + 15y) + ( 9x – 21y ) – (11x – 17y) = 19x + 11y
Determining polynomial functions - Example problem 2:
Simplify f(x) = (15x^2 – 11x – 21) + (7x^2 – 16x – 19) - (–7x^2 + 5x + 21)
Solution:
(15x^2 – 11x – 21) + (7x^2 – 16x – 19) - (–7x^2 + 5x + 21)
= 15x^2 – 11x – 21 + 7x^2 – 16x – 19 + 7x^2 - 5x - 21
= 15x^2 + 7x^2 + 7x^2 – 11x – 16x - 5x – 21 – 19 - 21
= 29x^2– 36x – 61
The answer for f(x) = (15x^2 – 11x – 21) + (7x^2 – 16x – 19) - (–7x^2 + 5x + 21) = 29x^2– 36x – 61
Determining polynomial functions - Example problem 3:
Simplify f(x) = (14x3 + 8x^2 + 21x – 16) – (18x3 – 15x^2 – 7x + 17) - (8x^2 – 11x – 21)
Solution:
(14x3 + 8x^2 + 21x – 16) – (18x3 – 15x^2 – 7x + 17) - (8x^2 – 11x – 21)
= (14x3 + 8x^2 + 21x – 16) – 1(18x3 – 15x^2 – 7x + 17) – 1(8x^2 – 11x – 21)
= (14x3 + 8x^2 + 21x – 16)– 1(18x3) – 1 (–15x^2) – 1(–7x) – 1(17) - 1(8x^2) -1(-11x) -1(-21)
= 14x3 + 8x^2 + 21x – 16 – 18x3 + 15x^2 + 7x – 17 - 8x^2 + 11x + 21
= 14x3 – 18x3 + 8x^2 + 15x^2 -8x^2 + 21x + 7x + 11x – 16– 17 + 21
= -4x3 + 15x^2 + 39x –20
The answer for f(x) = (14x3 + 8x^2 + 21x – 16) – (18x3 – 15x^2 – 7x + 17) - (8x^2 – 11x – 21) = -4x3 + 31x^2 + 39x –20
Practice Problems for Determining Polynomial Functions :
1. Simplify f(x)= (15x + 8y) + ( 7x – 15y ) – (8x – 9y)
The answer for f(x)= (15x + 8y) + ( 7x – 15y ) – (8x – 9y) = 14x + 2y
2. Simplify f(x)= (7x^2 – 9x – 8) + (3x^2 – 8x – 11) - (–6x^2 + 9x + 15)
The answer for f(x)=(7x^2 – 9x – 8) + (3x^2 – 8x – 11) - (–6x^2 + 9x + 15)= 16x^2– 26x – 34
In mathematics, a polynomial is an expression of finite length constructed from variables (also known as indeterminate) and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. For example, x2 - 4x + 7 is a polynomial, but x2 - 4/x + 7x3/2 is not, because its second term involves division by the variable x and because its third term contains an exponent that is not a whole number. (Source: Wikipedia)
Example Problems for Determining Polynomial Functions :
Determining polynomial functions - Example problem 1:
Simplify f(x) = (21x + 15y) + ( 9x – 21y ) – (11x – 17y)
Solution:
f(x) = (21x + 15y) + ( 9x – 21y ) – (11x – 17y) = 21x + 15y + 9x – 21y – 11x + 17y
= 21x + 9x - 11x + 15y – 21y + 17y
= 19x +11y
The answer for f(x) = (21x + 15y) + ( 9x – 21y ) – (11x – 17y) = 19x + 11y
Determining polynomial functions - Example problem 2:
Simplify f(x) = (15x^2 – 11x – 21) + (7x^2 – 16x – 19) - (–7x^2 + 5x + 21)
Solution:
(15x^2 – 11x – 21) + (7x^2 – 16x – 19) - (–7x^2 + 5x + 21)
= 15x^2 – 11x – 21 + 7x^2 – 16x – 19 + 7x^2 - 5x - 21
= 15x^2 + 7x^2 + 7x^2 – 11x – 16x - 5x – 21 – 19 - 21
= 29x^2– 36x – 61
The answer for f(x) = (15x^2 – 11x – 21) + (7x^2 – 16x – 19) - (–7x^2 + 5x + 21) = 29x^2– 36x – 61
Determining polynomial functions - Example problem 3:
Simplify f(x) = (14x3 + 8x^2 + 21x – 16) – (18x3 – 15x^2 – 7x + 17) - (8x^2 – 11x – 21)
Solution:
(14x3 + 8x^2 + 21x – 16) – (18x3 – 15x^2 – 7x + 17) - (8x^2 – 11x – 21)
= (14x3 + 8x^2 + 21x – 16) – 1(18x3 – 15x^2 – 7x + 17) – 1(8x^2 – 11x – 21)
= (14x3 + 8x^2 + 21x – 16)– 1(18x3) – 1 (–15x^2) – 1(–7x) – 1(17) - 1(8x^2) -1(-11x) -1(-21)
= 14x3 + 8x^2 + 21x – 16 – 18x3 + 15x^2 + 7x – 17 - 8x^2 + 11x + 21
= 14x3 – 18x3 + 8x^2 + 15x^2 -8x^2 + 21x + 7x + 11x – 16– 17 + 21
= -4x3 + 15x^2 + 39x –20
The answer for f(x) = (14x3 + 8x^2 + 21x – 16) – (18x3 – 15x^2 – 7x + 17) - (8x^2 – 11x – 21) = -4x3 + 31x^2 + 39x –20
Practice Problems for Determining Polynomial Functions :
1. Simplify f(x)= (15x + 8y) + ( 7x – 15y ) – (8x – 9y)
The answer for f(x)= (15x + 8y) + ( 7x – 15y ) – (8x – 9y) = 14x + 2y
2. Simplify f(x)= (7x^2 – 9x – 8) + (3x^2 – 8x – 11) - (–6x^2 + 9x + 15)
The answer for f(x)=(7x^2 – 9x – 8) + (3x^2 – 8x – 11) - (–6x^2 + 9x + 15)= 16x^2– 26x – 34
