- Polynomials
- General Form
- Polynomials: Parts
- Polynomial Graphs
- Like and Unlike Terms
- Degree of Polynomials
- Coefficient
- Polynomial Classification
- Polynomials in One Variable
- Degree of a Polynomial with One Variable
- Polynomials in Two Variable
- Adding Polynomials
- Subtracting Polynomials
- Multiplying Polynomials
- Polynomial Division Algorithm
- Steps for Polynomial Division
- Long Division Method

- Square roots of variables

- Negative exponents

- Fractional powers

- No variables in the denominator of any fraction

is called the leading coefficient.

is called the dominating term.

is called the constant term.

is the y-intercept of the graph of .

P(x) = 5x^{3
}+ 3x + 7

Type of Graph | Properties |

Smooth Graph | Rounded curves with no sharp corners. |

Continuous Graph | Graph has no breaks and can be drawn without lifting the pencil from the rectangular coordinate system. |

Like Terms | Unlike terms |

The like terms have same variable and same degree. | The unlike terms have different variables and different degrees. |

2x and 4x are the like terms. | 3x |

- When the variable does not have an exponent, it is assumed to be 1.

- 3x is the first degree polynomial.

- 3x
^{2 }is the second degree polynomial. 3x^{5 }- 4x + 3 is the fifth degree polynomial.

Term | Coefficient |

x | 1 |

-7x | -7 |

ax | a |

Type | Examples |

Monomial: Has one term | 5y or – 8x |

Binomial: Has two terms | -3x or 9y – 2y |

Trinomial: Has three terms | -3x or 9y |

__Classification of Polynomials Based on Degree:
__

Degree | Name | Example |

- | Zero | 0 |

0 | (Nonzero) constant | 1 |

1 | Linear | a + 1 |

2 | Quadratic | a |

3 | Cubic | a |

4 | Quartic(or biquadratic) | a |

5 | Quintic | a |

6 | Hexic | a |

7 | Septic | a |

8 | Octic | a |

9 | Nonic | a |

10 | Decic | a |

- n is a non-negative integer.

- x is a real number.

- a is called the coefficient of a term.

- Here we have one term with variable x and the exponent (or degree) as 1.

- Here we have two terms, with one variable x having the exponent (or degree) as 2 and another is constant value 2.

- This is a polynomial with one variable.

- The highest degree is 4.

- This is a polynomial with one variable.

- The highest degree is 2.

(5x + 7y) + (2x - y)

(5x + 2x) + (7y - y)

= (7x + 6y)

- 4x
^{2 }– 4 + x^{2 }+ 4x - 4

- = 4x
^{2 }+ x^{2 }+ 4x – 4 – 4

- = 5x
^{2 }+ 4x – 8

- Multiply each term in one polynomial by each term in the other polynomial.

- Add the resulting terms together, and simplify if needed.

= (x + 3)(x) + (x + 3)(2)

= x(x) + 3(x) + x(2) + 3(2)

= x^{2
}+ 3x + 2x + 6

= x^{2
}+ 5x + 6

- With d(x) = divisor and f(x) = dividend

- And the degree of d(x) is less than or equal to the degree of f(x).

- Then there exist unique polynomials q(x) and r(x) such that:

- Divide the first term in the dividend by the first term in the divisor.

- The result is the first term of the quotient.

- Write the resulting product beneath the dividend with like terms lined up.

- Bring down the next term in the original dividend and write it next to the remainder to form a new dividend.

- Multiply the x (on top) by the x (on the “side”), and carry the underneath.

- Then multiply the x (on top) by the 1 (on the “side”), and carry the 1x underneath.

- To subtract the polynomials, change all the signs in the second line and then add down.

- The first term () will cancel out,

- Now, carry down that last term -10,

- Divide the –10x by the x, it ends up with –10, so put that on top:

- Now multiply –10 by the leading x:

- Now multiply the –10 (on top) by the 1 (on the “side”), and carry the –10 to the bottom:

L.H.S. =

= R.H.S.

Hence proved L.H.S = R.H.S