A degree-6 polynomial can describe a roller-coaster track. A constant polynomial is just a flat line. Between those two extremes lies a rich family of expressions that power everything from graphics engines to engineering simulations. In this lesson you'll learn to identify, classify, and evaluate polynomials, the foundation of all that follows in Module 2.
Without using a definition is $f(x) = x^2 + \dfrac{1}{x}$ a polynomial? What about $g(x) = \sqrt{x} + 3$? Write your instinct and explain your reasoning before reading on.
There are only two core skills in this lesson, and everything else flows from them. Lock the polynomial definition (non-negative integer powers only) and direct substitution for evaluation into muscle memory.
Every polynomial question sits on one of two roads: identify whether an expression is a polynomial by checking powers, or evaluate the polynomial by substituting a value and simplifying.
A polynomial in $x$ is an expression of the form:
where $n$ is a non-negative integer (the degree), each $a_i$ is a constant coefficient, and $a_n \ne 0$ (so the degree is exactly $n$).
Examples of polynomials:
Not polynomials:
Polynomial: P(x) = a_n x^n + + a_0 where each power is a non-negative integer; Degree = highest power with a non-zero coefficient
Pause, copy the polynomial definition into your book: $P(x) = a_n x^n + \cdots + a_1 x + a_0$ where each power is a non-negative integer; degree = highest power with a non-zero coefficient.
Quick check: Which of the following is a polynomial?
We just saw that a polynomial $P(x) = a_n x^n + \cdots + a_0$ has degree $n$ (highest non-zero power). That raises a question: to check whether a given value $x = a$ satisfies $P(x) = 0$, you need to evaluate $P(a)$, what is the safest method to avoid sign errors? This card answers it → substitute using brackets around the value, evaluate each term separately, then combine.
To evaluate a polynomial at a specific value, substitute the value for $x$ and simplify step by step. Do not skip steps, sign errors are the main source of lost marks.
Example: If $P(x) = 2x^3 - x^2 + 3x - 4$, find $P(2)$.
You can also use evaluation to find unknown coefficients. If $P(x) = x^3 - 2x^2 + kx + 5$ and $P(2) = 11$:
Substitute x = a and evaluate each term separately before combining; Use brackets around substituted values: write 2(2)^3 not 22^3
Pause, copy the evaluation rule into your book: substitute using brackets, write $2(2)^3$ not $22^3$; evaluate each term separately before combining to avoid sign errors.
Did you get this? True or false: if $P(x) = 3x^2 - x + 2$, then $P(-1) = 6$.
Worked examples · 3 in a row, reveal as you go
Determine the degree and leading coefficient of $P(x) = 5x^4 - 2x^6 + 3x - 1$.
If $P(x) = 2x^3 - x^2 + 3x - 4$, find $P(-2)$.
Find the value of $k$ if $P(x) = x^3 - 2x^2 + kx + 5$ has $P(2) = 11$.
Fill the gap: If $P(x) = x^3 - 4x + 1$, then $P(2) =$ .
Misconceptions to fix · the 3 traps that cost marks
Did you get this? True or false: $f(x) = x^3 + \sqrt{x}$ is a polynomial.
Activities · practice with the ideas
State whether each expression is a polynomial. If not, explain why: (a) $4x^3 - 7x + 2$ (b) $x^{-2} + 3x$ (c) $6$ (d) $\dfrac{x^2 + 1}{x}$
Find the degree and leading coefficient of $Q(x) = 7 - 3x^5 + x^2 - 9x^3$.
If $P(x) = 2x^2 - 3x + k$ and $P(1) = 4$, find $k$.
Evaluate $P(x) = x^4 - 3x^2 + 2x - 5$ at $x = -1$.
Explain in your own words why every constant (e.g. $P(x) = 7$) qualifies as a polynomial, and what its degree is.
Odd one out: Which expression does NOT belong with the others? Select and explain.
Earlier you were asked whether $f(x) = x^2 + \dfrac{1}{x}$ and $g(x) = \sqrt{x} + 3$ are polynomials.
$f(x) = x^2 + x^{-1}$, the $x^{-1}$ term has a negative power, so $f$ is not a polynomial. $g(x) = x^{1/2} + 3$, the $x^{1/2}$ term has a fractional power, so $g$ is also not a polynomial. A polynomial requires all powers to be non-negative integers.
Pick your answer, then rate your confidence. That tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. State the degree and leading coefficient of $P(x) = 3x^5 - 2x^3 + x - 7$. (2 marks)
Q2. If $P(x) = 2x^2 - 3x + k$ and $P(1) = 4$, find $k$. (2 marks)
Q3. Explain why $f(x) = x^{-2} + 3x$ is not a polynomial. What would need to change to make it a polynomial? (2 marks)
Activity 1: (a) Polynomial, all non-negative integer powers. (b) Not a polynomial, $x^{-2}$ has a negative power. (c) Polynomial, constant, degree 0. (d) Not a polynomial, $\frac{x^2+1}{x} = x + x^{-1}$, contains $x^{-1}$.
Activity 2: Rewrite: $-3x^5 + x^2 - 9x^3 + 7 = -3x^5 - 9x^3 + x^2 + 7$. Degree = 5; leading coefficient = $-3$.
Activity 3: $P(1) = 2 - 3 + k = k - 1 = 4 \Rightarrow k = 5$.
Activity 4: $P(-1) = (-1)^4 - 3(-1)^2 + 2(-1) - 5 = 1 - 3 - 2 - 5 = -9$.
Activity 5: $7 = 7 \cdot x^0$, it fits the form $a_0 x^0$ with non-negative integer power 0. Degree = 0.
Q1 (2 marks): Degree = 5 [1]; leading coefficient = 3 [1].
Q2 (2 marks): $P(1) = 2(1)^2 - 3(1) + k = 2 - 3 + k = k - 1$ [1]. Set $k - 1 = 4$, so $k = 5$ [1].
Q3 (2 marks): $f(x) = x^{-2} + 3x$ contains the term $x^{-2}$, which has a negative integer exponent $(-2)$. Polynomials require all exponents to be non-negative integers [1]. To make it a polynomial, replace $x^{-2}$ with a term like $x^2$ or any $x^n$ where $n \geq 0$ [1].
A full module quiz covering every lesson in this module, not just this one. Set aside a decent block of time and treat it like a real assessment.
Start the module quiz →Tick when you've finished the practice and review.
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