**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 *x ^{n}* 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):**

** (2x ^{2}y)(3x^{4}y^{2})**

**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.

** (x ^{2} * x^{4}) à (x^{2}+4=x^{6})**

**Step 3:** Multiply the variable with the base of y. If the bases are same add exponent.

**(y1 * y ^{2}) à (y1+2= y^{3})**

** The Final answer is 6x ^{6}y^{3}**

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## Problems in polynomials

**Solving Polynomials problems:**

**Example 1: **

Simplify **(7x ^{2} – x – 5) + (x^{2} – 2x – 4) + (–2x^{2 }+ 3x + 6) (concept using Polynomials)**

** (7x ^{2} – x – 5) + (x^{2} – 2x – 4) + (–2x^{2} + 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,

**= 7x ^{2} – x – 5 + x^{2} – 2x – 4 + –2x^{2} + 3x +6**

** = 7x ^{2} + 1x^{2} – 2x^{2} – 1x – 2x + 3x – 5 – 4 + 6**

** = 8x ^{2} – 2x^{2} – 3x + 3x – 9 +6**

** = 6x ^{2} – 3**

** Example 2:**

** (3x ^{3} + 3x^{2 }– 4x + 5) + (x^{3} – 2x^{2} + x – 4) (concept using Polynomials)**

** Solution:**

**Step 1:** (3x^{3} + 3x^{2} – 4x + 5) + (x^{3 }– 2x^{2} + x – 4).

**Step 2:** It can be written as 3x^{3} + 3x^{2} – 4x + 5 + x^{3} – 2x^{2} + x – 4.

**Step 3:** now we need to simplify this, so.

**Step 4:** 3x^{3} + x^{3} + 3x^{2} – 2x^{2} – 4x + x + 5 – 4.

**Step 5**: So, the answer is **4x ^{3} + 1x^{2 }– 3x + 1.**