What is a polynomial?
A polynomial is a mathematical expression with many terms ("poly" means "many" and "nomial" means "names" or "terms").
A polynomial can have any combination of operators (addition, subtraction, multiplication or division), constants, variables and exponents, but never division by a variable. Remember, this means that expressions with negative exponents can never be polynomials because $a^{-x}$a−x is the same as $\frac{1}{a^x}$1ax. Click here if you need a refresher about negative exponents.
Mathematicians like order so polynomials are usually written in descending order, starting with the term with the highest power and ending with the term with the lowest (or no) power. For example, in the polynomial $8x+4x^8-2x^2+4$8x+4x8−2x2+4, the powers are all jumbled. To put it in order, we would rewrite it as $4x^8-2x^2+8x+4$4x8−2x2+8x+4.
| Examples of expressions that ARE polynomials | Examples of expressions that ARE NOT polynomials |
|---|---|
| $5x^2+\frac{4}{3}x-7$5x2+43x−7 | $\frac{4}{x-3}$4x−3 |
| $-18$−18 | $3+\frac{1}{x}$3+1x |
| $3x$3x | $4x^3-\frac{1}{x^7}+8$4x3−1x7+8 |
| $4c-8cd+2$4c−8cd+2 | $\frac{7}{8}x^{-2}+5$78x−2+5 |
| $22x^6+12y^8$22x6+12y8 | $\sqrt{x}$√x |
| $7g+\sqrt{12}$7g+√12 | $12f^3g^{-4}\times h^6$12f3g−4×h6 |
If an expression contains terms that divide by a variable (i.e. an algebraic term), they are NOT polynomials.
Parts of a polynomial
Degree: The largest exponent (i.e. power) of a variable in a polynomial. e.g. In the polynomial $x^3+4x^2-9$x3+4x2−9, the highest power of $x$x is $3$3, so the degree in this polynomial is $3$3.
Leading coefficient: When a polynomial is written with its exponents in descending order, the leading coefficient is the number that is written before the first algebraic term. For example, in $5x-7$5x−7, the leading coefficient is $5$5. Sometimes you may need to use your knowledge of algebra to work out the leading coefficient. e.g. In $-x^5-2x^4+4$−x5−2x4+4, the leading coefficient is $-1$−1.
Constant term: the term in a polynomial that has no variables (i.e. no algebraic terms). e.g. in the polynomial $4y^8+2xy-4x-\frac{2}{3}$4y8+2xy−4x−23, the constant term is $-\frac{2}{3}$−23.
Working with polynomials
Since polynomials often have quite a few terms, we use function notation to write out mathematical expressions involving polynomials. When we want to work mathematically with different polynomials, whether it be to simplify or evaluate them, we need to use our knowledge of simplifying algebraic terms and exponent laws to do so.
Tip: If you're working with two functions in a question, it can be helpful to write the functions in brackets to start to make sure you end up with all the right signs.
Examples
If $P(x)=-3x^2-6x+6$P(x)=−3x2−6x+6 and $Q(x)=2x+7$Q(x)=2x+7:
A) Find $P(x)+Q(x)$P(x)+Q(x)
Think: We need to substitute in the values of $x$x.
Do:
| $-3x^2-6x+6+2x+7$−3x2−6x+6+2x+7 | $=$= | $-3x^2-6x+6+2x+7$−3x2−6x+6+2x+7 | (Let's group the like terms and simplify) |
| $=$= | $-3x^2-4x+13$−3x2−4x+13 |
B) Find $P(x)-Q(x)$P(x)−Q(x)
| $-3x^2-6x+6-\left(2x+7\right)$−3x2−6x+6−(2x+7) | $=$= | $-3x^2-6x+6-2x-7$−3x2−6x+6−2x−7 |
| $=$= | $-3x^2-8x-1$−3x2−8x−1 |
C) Find $P(x)\times Q(x)$P(x)×Q(x)
| $P(x)\times Q(x)$P(x)×Q(x) | $=$= | $\left(-3x^2-6x+6\right)\times\left(2x+7\right)$(−3x2−6x+6)×(2x+7) | (Let's expand the brackets using binomial expansion) |
| $=$= | $-6x^3-12x^2+12x-21x^2-42x+42$−6x3−12x2+12x−21x2−42x+42 | ||
| $=$= | $-6x^3-33x^2-30x+42$−6x3−33x2−30x+42 |
D) Find $P(-x)$P(−x)
Think: This means that wherever $x$x is written in the equation, we are going to sub in $-x$−x.
Do: $-3\times\left(-x\right)^2-6\times\left(-x\right)+6=-3x^2+6x+6$−3×(−x)2−6×(−x)+6=−3x2+6x+6
More Worked Examples
QUESTION 1
Is $2x^3-4x^5+3$2x3−4x5+3 a polynomial?
Yes, it is a polynomial.
ANo, it is not a polynomial.
B
QUESTION 2
Consider $P\left(x\right)=4x^5+3x^6-8$P(x)=4x5+3x6−8
Find the value of $P$P$\left(0\right)$(0).
Find the value of $P$P$\left(1\right)$(1).
Find the value of $P$P$\left(-4\right)$(−4).
QUESTION 3
If $P\left(x\right)$P(x)$=$=$-5x^2-6$−5x2−6, $Q\left(x\right)=x+1$Q(x)=x+1, and $R\left(x\right)=5x^2+3x$R(x)=5x2+3x
Find $P\left(x\right)+Q\left(x\right)+R\left(x\right)$P(x)+Q(x)+R(x)
Find $Q\left(x\right)-P\left(x\right)$Q(x)−P(x)
Find $P\left(x\right)\times R\left(x\right)$P(x)×R(x).
Find $P\left(x\right)-Q\left(x\right)-R\left(x\right)$P(x)−Q(x)−R(x)
Find $R\left(x\right)-Q\left(x\right)P\left(x\right)$R(x)−Q(x)P(x)