Table of Contents

**What is Polynomial?**

Poly means “many” and nomials means “terms”. Polynomial means one of algebraic expression that consist of variable, coefficient, and constant.

The characteristic of polynomial is it has non-negative integer power. Polynomial can be written as

*a*_{n}**x**^{n}** + ***a*_{n-1}**x**^{n-1}** + ***a*_{n-2}**x**^{n-2 }**+ …. + ***a*_{2}**x**^{2}** + ***a*_{1}**x**^{1}** + ***a*_{0}

or

Where

*a*_{n},….*a*_{0} : any number(as coefficient and constant)

x : variable,

n, n-1, .. : non-negative integer power (it is arranged from the highest power)

More specific, there are some types of polynomial.

**Monomial**

Mono means “one”. Monomial means has one term. It should be non-zero term. For example: 2x, 3y^{2}, -4, etc.

**Binomial**

Binomial has two terms. There is subtraction or addition between the terms. For example: 2x-1, x^{3}-2, -y^{4}+4.

**Trinomial**

Trinomial means polynomial that consist of tri terms. There are subtraction or addition between the terms. For example: 2x^{3}-x-1, x^{5}-x^{2}+3

Generally, polynomial can be operated wit addition, subtraction, multiplication, and division (but no division by a variable).

**Adding Polynomials**

Adding polynomials is like adding integer number. But the important thing is be careful with the variable and degree (power).

The main concept of adding polynomial is adding the terms that has same variable and degree.

It can not be operated if the terms have different variable or different power. It will be easier if you collect the terms that has same variable and degree and operated it.

If there is polynomial (2x^{4 }– 4x^{3} + 1) add with (y^{4}), can it be solved?

The answer is NO.** **

Why?

because the both of polynomials has different variable. So, the solution is only 2x^{4 }+ y^{4 }– 4x^{3} +1.

But how if (2x^{4 }– 4x^{3} + 1) add with (x^{4})?

The answer is 2x^{4 }+ x^{4 }– 4x^{3} + 1 = (2+1) x^{4} – 4x^{3} +1 = 3x^{4} -4x^{3} +1

**Subtracting Polynomials**

Subtracting polynomials has same concept with adding polynomials. The important things are variable and degree. Another thing that need your carefulness is parentheses. It will differently value if there are parentheses or not between the terms. It will be easier if you collect the terms that has same variable and degree and operated it.

If there is polynomial (3x^{5} + 2x^{3} – x^{2} + 1) that is subtracted from (2x^{3} – 2x^{2 }+ 3), how the answer?

The answer is

2x^{3} – 2x^{2 }+ 3 – (3x^{5} + 2x^{3} – x^{2} + 1) = 2x^{3} – 2x^{2 }+ 3 – 3x^{5} – 2x^{3} + x^{2} -1

(be careful with the sign and the parentheses)

= -3x^{5} + 2x^{3} – 2x^{3 }+ x^{2 }– 2x^{2 }+ 3 – 1

(be careful when you collect the terms based on degree. Making sure that all of terms has been written)

= -3x^{5 }– x^{2 }+ 2

(here is the answer)

**Polynomial Examples**

- Which one that is polynomial.

a. | 2x | e. | 3 |

b. | x^{3}– ½ | f. | x^{4}-5 |

c. | x-2 | g. | √x + x^{2} |

d. | 2/(x-1) | h. | 2xy^{3}z + xyz – 4 |

Polynomial:

a. | 2x (equals with 2x^{1}) |

b. | x^{3}– ½ (non-negative integer power) |

c. | x-2 (non-negative integer power) |

e. | 3 (equals with 3x^{0}) |

f. | x^{4}-5 (non-negative integer power) |

h. | 2xy^{3}z + xyz – 4 (variable is only y with non-negative integer power) |

Non-Polynomial:

d. | 2/(x-1) (polynomial cannot divide by variable) |

g. | √x + x^{2 }(equals with x^{1/2}+x^{2}, fraction power) |

2. Subtract (5x^{6} – 2x^{4} + x^{2} + 1) from (3x^{4 }+ x^{2 }– 8)

3x^{4 }+ x^{2 }– 8 – (5x^{6} – 2x^{4} + x^{2} + 1) = 3x^{4 }+ x^{2 }– 8 – 5x^{6} + 2x^{4} – x^{2} – 1

= – 5x^{6} + 3x^{4} + 2x^{4} + x^{2} – x^{2} – 8 – 1

= – 5x^{6} + 5x^{4} – 9

3. If A = 3x^{2} – 2x, B = 3x^{6} – x^{3}, and C = x^{2} + 4, determine the polynomial of A+B-C.

A + B – C = 3x^{2} – 2x + 3x^{6} – x^{3} – x^{2} + 4

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

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