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 integer exponents, but never division by a variable. This means that expressions with negative indices can never be polynomials because $x^{-a}$x−a is the same as $\frac{1}{x^a}$1xa.
Examples of expressions that ARE polynomials | Examples of expressions that ARE NOT polynomials |
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$5x^2+\frac{4}{3}x-7$5x2+43x−7 | $\frac{4}{x-3}$4x−3 |
$-18$−18 | $4x^3-\frac{1}{x^7}+8$4x3−1x7+8 |
$7g+\sqrt{12}$7g+√12 | $\frac{7}{8}x^{-2}+5$78x−2+5 |
$22x^6+12y^8$22x6+12y8 | $\sqrt{x}$√x |
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. When the leading coefficient is 1, the polynomial is called a monic polynomial.
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.
Is $2x^3-4x^5+3$2x3−4x5+3 a polynomial?
Yes, it is a polynomial.
No, it is not a polynomial.
For the polynomial $P\left(x\right)=\frac{x^7}{5}+\frac{x^6}{6}+5$P(x)=x75+x66+5
The degree of the polynomial is: $\editable{}$
The leading coefficient of the polynomial is: $\editable{}$
The constant term of the polynomial is: $\editable{}$
If we start with the most basic form of a polynomial, $y=ax^n$y=axn, we can immediately see a pattern emerge for odd and even powers. Experiment with the following applet, where $a=1$a=1.
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Odd degree polynomials move in opposite directions at the extremities and even degree polynomials move in the same direction at the extremities! This is because the degree of a polynomial, $n$n, together with the leading coefficient, $a$a, dictate the overall shape and behaviour of the function at the extremities:
$n$n | $a>0$a>0 | $a<0$a<0 |
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Even |
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Odd |
But what happens between the extremities?
We know that a polynomial of degree $n$n can have up to $n$n $x$x-intercepts, with those of odd degree having at least one.
A polynomial of degree $n$n can have up to $n-1$n−1 turning points, with those of even degree having at least one.
Details of key features can be found using Calculus and technology or from factorised forms of the function. Let's focus on what we can learn from factorised forms of polynomials.
Certain polynomials of degree $n$n can be factorised into $n$n linear factors over the real number field. For example the $4$4th degree polynomial $P\left(x\right)=2x^4-x^3-17x^2+16x+12$P(x)=2x4−x3−17x2+16x+12 can be re-expressed as $P\left(x\right)=\left(x+3\right)\left(2x+1\right)\left(x-2\right)^2$P(x)=(x+3)(2x+1)(x−2)2. Note that there are two distinct factors, the $\left(x+3\right)$(x+3) and $\left(2x+1\right)$(2x+1) and two equal factors of $\left(x-2\right)$(x−2) . We can immediately identify that there are roots at $x=-3,-\frac{1}{2}$x=−3,−12 and $x=2$x=2.
The function $y=-\left(x-1\right)\left(x+2\right)^2$y=−(x−1)(x+2)2 has a root at $x=1$x=1 and a repeated root at $x=-2$x=−2. The $y$y - intercept is $-\left(-1\right)\left(2\right)^2=4$−(−1)(2)2=4. The function is of odd degree, so its ends move off in different directions. Note that for large positive values of $x$x, the curve becomes very negative (See graph $G3$G3 below).
Which of the following is the graph of the function $f\left(x\right)=x\left(x+3\right)\left(x-3\right)$f(x)=x(x+3)(x−3)?
Consider the function $y=-\left(x-2\right)^2\left(x+1\right)$y=−(x−2)2(x+1).
Find the $x$x-value(s) of the $x$x-intercept(s).
Write each line of working as an equation. If there is more than one answer, write all solutions on the same line separated by commas.
Find the $y$y-value of the $y$y-intercept.
Write each line of working as an equation.
Which of the following is the graph of $y=-\left(x-2\right)^2\left(x+1\right)$y=−(x−2)2(x+1)?
Consider the function $y=x^4-x^2$y=x4−x2
Determine the leading coefficient of the polynomial function.
Is the degree of the polynomial odd or even?
odd
even
Which of the following is true of the graph of the function?
It rises to the left and rises to the right
It rises to the left and falls to the right
It falls to the left and rises to the right
It falls to the left and falls to the right
Which of the following is the graph of $y=x^4-x^2$y=x4−x2?