12.01 Polynomials
An expression of the form $Ax^n$Axn, where $A$A is any number and $n$n is any non-negative integer, is called a monomial. When we take the sum of multiple monomials, we get a polynomial.
Parts of polynomials
In the monomial $Ax^n$Axn:
- $A$A is the coefficient
- $x$x is the variable
- $n$n is the index
A polynomial is a sum of any number of monomials (and consequently, each term of a polynomial is a monomial). The highest index is called the degree of the polynomial. For example, $x^3+4x+3$x3+4x+3 is a polynomial of degree three. The coefficient of the term with the highest index is called the leading coefficient. The coefficient of the term with index $0$0 is called the constant.
We often name polynomials using function notation. For example $P\left(x\right)$P(x) is a polynomial where $x$x is the variable. If we write a constant instead of $x$x, that means that we substitute that constant for the variable. For example, if $P\left(x\right)=x^3+4x+3$P(x)=x3+4x+3 then $P\left(3\right)=3^3+4\times3+3=42$P(3)=33+4×3+3=42.
Polynomials of particular degrees are given specific names. Some of these we have seen before.
| Degree | Name |
|---|---|
| Zero | Constant |
| One | Linear |
| Two | Quadratic |
| Three | Cubic |
Operations on polynomials
We apply operations to polynomials in the same way as we apply operations to numbers. For addition and subtraction we add or subtract all of the terms in both polynomials and we simplify by collecting like terms. For multiplication we multiply each term in one polynomial by each term in the other polynomial similar to how we expand binomial products. Division is a more complicated case that we will look at in the next chapter.
Worked example
$P\left(x\right)=x^3+4x+3$P(x)=x3+4x+3 and $Q\left(x\right)=4x^2-6x-7$Q(x)=4x2−6x−7. Find $P\left(x\right)+Q\left(x\right)$P(x)+Q(x), $P\left(x\right)-Q\left(x\right)$P(x)−Q(x) and $P\left(x\right)\times Q\left(x\right)$P(x)×Q(x).
Think: For each of these we start by substituting the polynomials and then we can use the algebraic techniques we've already learned to simplify them.
Do:
| $P\left(x\right)+Q\left(x\right)$P(x)+Q(x) | $=$= | $\left(x^3+4x+3\right)+\left(4x^2-6x-7\right)$(x3+4x+3)+(4x2−6x−7) |
Substituting expressions for $P\left(x\right)$P(x) and $Q\left(x\right)$Q(x) |
| $=$= | $x^3+4x^2-2x-4$x3+4x2−2x−4 |
Gathering like terms and simplifying |
|
| $P\left(x\right)-Q\left(x\right)$P(x)−Q(x) | $=$= | $\left(x^3+4x+3\right)-\left(4x^2-6x-7\right)$(x3+4x+3)−(4x2−6x−7) |
Substituting expressions for $P\left(x\right)$P(x) and $Q\left(x\right)$Q(x) |
| $=$= | $x^3+4x+3-4x^2+6x+7$x3+4x+3−4x2+6x+7 |
Expanding the brackets |
|
| $=$= | $x^3-4x^2+10x+10$x3−4x2+10x+10 |
Gathering like terms and simplifying |
|
| $P\left(x\right)\times Q\left(x\right)$P(x)×Q(x) | $=$= | $\left(x^3+4x+3\right)\left(4x^2-6x-7\right)$(x3+4x+3)(4x2−6x−7) |
Substituting expressions for $P\left(x\right)$P(x) and $Q\left(x\right)$Q(x) |
| $=$= | $4x^5-6x^4-7x^3+16x^3-24x^2-28x+12x^2-18x-21$4x5−6x4−7x3+16x3−24x2−28x+12x2−18x−21 |
Expanding the brackets. Note that every term in $P\left(x\right)$P(x) is multiplied by every term in $Q\left(x\right)$Q(x). Since there are three terms in each polynomial, there are nine terms here. |
|
| $=$= | $4x^5-6x^4+9x^3-12x^2-46x-21$4x5−6x4+9x3−12x2−46x−21 |
Gathering like terms and simplifying |
A monomial is an expression of the form $Ax^n$Axn. In this monomial:
- $A$A is the coefficient
- $x$x is the variable
- $n$n is the index
A polynomial is a sum of any number of monomials. In a polynomial:
- The highest index is the degree
- The coefficient of the term with the highest index is the leading coefficient
- The coefficient of the term with index $0$0 is the constant
We apply operations to polynomials in the same way that we apply operations to numbers.
Practice questions
Question 1
For the polynomial $P\left(x\right)=5x^4+7x^2+5x+3$P(x)=5x4+7x2+5x+3
What's the degree of the polynomial? $\editable{}$
What's the leading coefficient of the polynomial? $\editable{}$
What's the constant term of the polynomial? $\editable{}$
Question 2
For the polynomial $P\left(x\right)=4-x^2$P(x)=4−x2
Find $P\left(-1\right)$P(−1).
Find $P\left(\sqrt{2}\right)$P(√2).
Find $P\left(\frac{1}{2}\right)$P(12).
Question 3
Simplify $\left(-4x^3-4x^2+1\right)+\left(-x^2-9x-5\right)$(−4x3−4x2+1)+(−x2−9x−5).