Algebra 1 - Polynomials and Operations

Introduction

• Polynomials are mathematical expressions with one or more terms separated by an addition or subtraction operator.
• In a polynomial, the variables are raised to a non-negative integer power.
• For example, $x^2+7x+10$ is a polynomial in the variable $x$ with three terms.
• The highest power of the variable in any polynomial is called its degree.
• In the above example, the degree of the polynomial is two.
• Note: A polynomial with a single term is called a Monomial.

Addition and Subtraction of Polynomials

• The addition & subtraction of polynomials can be carried out in the following two steps:

• • Identify like terms in the polynomial
  • Combine them according to the correct integer operation (addition or subtraction)

• Note that like terms have the same variables raised to the same exponents.
• The addition or subtraction of polynomials results in a polynomial of the same degree.

Polynomial Multiplication

• To multiply a polynomial by a monomial, apply the distributive property and simplify each term.
• For example: $2x·\left(x-5\right)$$=2x·x-2x·5$$=2x^2-10x$
• Likewise, to multiply two polynomials, we distribute each term of the one polynomial to all the terms of the other polynomial (FOIL method).
• For example: $\left(5x-2\right)·\left(x+3\right)$$=5x\left(x+3\right)-2\left(x+3\right)$$=5x^2+15x-2x-6$$=5x^2+13x-6$
• Note: Two or more polynomials, when multiplied, always result in a polynomial of a higher degree (unless one of them is a constant polynomial).

Dividing Polynomials

• To divide a monomial by a monomial, divide the numerical coefficients, divide the variables separately, and then multiply the results.
• Example: $\frac{14x^6}{7x^2}$$=\frac{14}{7}·\frac{x^6}{x^2}$$=2x^4$
• To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.
• Example: $\frac{8m^5+6m^3-3m^2}{-2m}$$=\frac{8m^5}{-2m}+\frac{6m^3}{-2m}-\frac{3m^2}{-2m}$$=-4m^4-3m^2+\frac{3m}{2}$
• The long division method is used to divide a polynomial by a polynomial.

Solved Examples

Question 1: Add the following polynomials: $\left(5m^3-n^2\right)+\left(2m^3-6n^2\right)$.

Solution: $\left(5m^3-n^2\right)+\left(2m^3-6n^2\right)$$=5m^3-n^2+2m^3-6n^2$$=5m^3+2m^3-6n^2-n^2$$=7m^3-7n^2$

Question 2: Find the product: $\left(m-5\right)·\left(3m+6\right)$.

Solution: $\left(m-5\right)·\left(3m+6\right)$$=m·\left(3m+6\right)-5·\left(3m+6\right)$$=3m^2+6m-15m-30$$=3m^2-9m-30$

Question 3: Find the product: ${\left(5-x\right)}^2$.

Solution: ${\left(5-x\right)}^2$$=\left(5-x\right)·\left(5-x\right)$$=5·\left(5-x\right)-x\left(5-x\right)$$=25-5x-5x+x^2$$=25-10x+x^2$

Question 4: Divide $\left(-18x^7+4x^5-2x^2\right)$ by $x^2$.

Solution: $\frac{-18x^7+4x^5-2x^2}{x^2}$$=\frac{-18x^7}{x^2}+\frac{4x^5}{x^2}-\frac{-2x^2}{x^2}$$=-18x^5+4x^3+2$

Question 5: Find the degree of the polynomial $x^3-2x^2+4$.

Solution: The highest power of the variable in the given polynomial is three; hence, its degree is three.

Cheat Sheet

• To add or subtract the polynomials, add or subtract the like terms.
• To multiply two polynomials, multiply each term in the first polynomial by each term in the second polynomial.
• The "FOIL" method is used to multiply two binomials.
• To divide a polynomial with a monomial, divide each polynomial term by the monomial.
• To divide two polynomials, use the long division method.
• Below are some special product rules:

• • ${\left(x+y\right)}^2=x^2+2xy+y^2$
  • ${\left(x-y\right)}^2=x^2-2xy+y^2$
  • $\left(x+y\right)\left(x-y\right)=x^2-y^2$
  • ${\left(x+y\right)}^3=x^3+3x^{2}y+3x{y}^2+y^3$
  • ${\left(x-y\right)}^3=x^3-3x^{2}y+3x{y}^2-y^3$
  • $(x+y)(x^2-xy+y^2)=x^3+y^3$
  • $(x-y)(x^2+xy+y^2)=x^3-y^3$

Blunder Areas

• Only like (similar) terms of polynomials can be added or subtracted.
• One must be careful while multiplying polynomials involving negative signs.