Introduction to monomial and polynomials:
Monomials
The monomial is a product of powers of variables, or formally any value obtained from 1 by finitely many multiplications by a variable. If only a single variable x is to be considered this means that any monomial is either 1 or a power xn of x, with n a positive integer.
Polynomials
In mathematics, a polynomials is an expression of the 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.(Source in Wikipedia ).
Problems in Monomials:
Solving Monomials Examples:
Example 1:
Simplify the lowest number and find out the GCF. (Using the Concept of monomials)
96y, 12x, -8y
Solution:
96y, 12x, -8y
Factor out the GCF of 4 from each term,
4(24y) + 4(3x) + 4(-2y)
= 4(24y + 3x – 2y)
GCF = 4
Example 2:
Solving Problems on multiplying the monomials. (Using the Concept of monomials):
(2x2y)(3x4y2)
Solution:
Step 1: First Multiply the Coefficient first
2 * 3 = 6
Step 2: Multiply the variable with the base of x. If the bases are same add exponent.
(x2 * x4) à (x2+4=x6)
Step 3: Multiply the variable with the base of y. If the bases are same add exponent.
(y1 * y2) à (y1+2= y3)
The Final answer is 6x6y3
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Problems in polynomials
Solving Polynomials problems:
Example 1:
Simplify (7x2 – x – 5) + (x2 – 2x – 4) + (–2x2 + 3x + 6) (concept using Polynomials)
(7x2 – x – 5) + (x2 – 2x – 4) + (–2x2 + 3x + 6)
After clearing the parentheses and then we want to group the like terms and simplify the given polynomial which is done on the basis of coefficients of the terms,
= 7x2 – x – 5 + x2 – 2x – 4 + –2x2 + 3x +6
= 7x2 + 1x2 – 2x2 – 1x – 2x + 3x – 5 – 4 + 6
= 8x2 – 2x2 – 3x + 3x – 9 +6
= 6x2 – 3
Example 2:
(3x3 + 3x2 – 4x + 5) + (x3 – 2x2 + x – 4) (concept using Polynomials)
Solution:
Step 1: (3x3 + 3x2 – 4x + 5) + (x3 – 2x2 + x – 4).
Step 2: It can be written as 3x3 + 3x2 – 4x + 5 + x3 – 2x2 + x – 4.
Step 3: now we need to simplify this, so.
Step 4: 3x3 + x3 + 3x2 – 2x2 – 4x + x + 5 – 4.
Step 5: So, the answer is 4x3 + 1x2 – 3x + 1.
