407 493 6601

# 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)$$=2x·x-2x·5$$=2{x}^{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)$$=5x\left(x+3\right)-2\left(x+3\right)$$=5{x}^{2}+15x-2x-6$$=5{x}^{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{14{x}^{6}}{7{x}^{2}}$$=\frac{14}{7}·\frac{{x}^{6}}{{x}^{2}}$$=2{x}^{4}$
• To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.
• Example: $\frac{8{m}^{5}+6{m}^{3}-3{m}^{2}}{-2m}$$=\frac{8{m}^{5}}{-2m}+\frac{6{m}^{3}}{-2m}-\frac{3{m}^{2}}{-2m}$$=-4{m}^{4}-3{m}^{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: $\left(5{m}^{3}-{n}^{2}\right)+\left(2{m}^{3}-6{n}^{2}\right)$.

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

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

Solution: $\left(m-5\right)·\left(3m+6\right)$$=m·\left(3m+6\right)-5·\left(3m+6\right)$$=3{m}^{2}+6m-15m-30$$=3{m}^{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 $\left(-18{x}^{7}+4{x}^{5}-2{x}^{2}\right)$ by ${x}^{2}$.

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

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

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

## Blunder Areas

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