Introducing Polynomials
Every question with a fully worked solution.
- Introducing Polynomials - Video - Polynomials - Classifying Monomials, Binomials & Trinomials - Degree & Leading Coefficient Watch
Theory
A polynomial is a sum of terms \(ax^n\) where every power \(n\) is a whole number. This Year 10A maths page covers the degree, leading coefficient, constant term and monic polynomials, \(P(x)\) notation, operations on polynomials and finding unknown coefficients.
A polynomial in \(x\) has the form \(a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\), where the powers are whole numbers \(0, 1, 2, \ldots\) and the coefficients \(a_n, \ldots, a_0\) are real numbers.
The degree is the highest power of \(x\). The term with that power is the leading term, and its coefficient is the leading coefficient. A polynomial with leading coefficient \(1\) is monic. The constant term \(a_0\) has no \(x\).
Degrees \(1\), \(2\), \(3\) and \(4\) are called linear, quadratic, cubic and quartic.
\(P(x)\) notation names a polynomial; \(P(c)\) is its value when \(x = c\).
This prepares you for Mathematical Methods and Specialist Mathematics in Years 11 and 12.
| To find | Use |
|---|---|
| constant term | \(P(0)\) |
| sum of the coefficients | \(P(1)\) |
| leading term of a product | multiply the leading terms only |
| degree of \(P(x)\,Q(x)\) | \(\deg P + \deg Q\) |
| degree of \([P(x)]^k\) | \(k \times \deg P\) |
| degree of \(P(x) + Q(x)\) | at most the larger degree (equal unless the leading terms cancel) |
How to describe a polynomial
- Check every power of \(x\) is a whole number. Rewrite \(\dfrac1x\), \(\sqrt x\) and roots as powers first. A pronumeral in the index (\(2^x\)) is not allowed.
- Expand or simplify if needed, at least enough to find the highest power. Like terms may cancel: \(5x^3 - 5x^3 + x\) has degree \(1\).
- Read off the degree, leading coefficient and constant term. Monic means the leading coefficient is exactly \(1\).
Working with P(x)
- Substitute with brackets around negative numbers: \(-3(-2)^2 = -12\).
- Add or subtract by collecting like terms. Multiply out \(2Q(x)\) before subtracting, and change every sign of the polynomial being subtracted.
- Unknown coefficients: each given value \(P(c) = k\) gives one equation. Solve the equations simultaneously; \(P(0)\) often gives the constant straight away.
- Models: build the polynomial from the situation (for example volume \(= x(40 - 2x)(25 - 2x)\) for an open box), expand, then substitute.
(i) \(4 - x^3 + 2x^5\)
(ii) \((1 - 2x)^2(x + 4)\)
(iii) \(x + \dfrac{3}{x^2}\)
(i) Rearrange in descending powers: \(2x^5 - x^3 + 4\).
| \(\text{degree}\) | \(=\) | \(\mathbf{5}\) |
| \(\text{leading coefficient}\) | \(=\) | \(\mathbf{2}\) |
| \(\text{constant term}\) | \(=\) | \(\mathbf{4}\) |
The leading coefficient is \(2\), not \(1\), so it is not monic.
(ii) No need to expand fully. Multiply the highest-power terms for the leading term, and substitute \(x = 0\) for the constant.
| \(\text{leading term}\) | \(=\) | \((-2x)^2 \times x = 4x^3\) |
| \(\text{constant}\) | \(=\) | \((1 - 0)^2(0 + 4) = 4\) |
Degree \(\mathbf{3}\) (a cubic), leading coefficient \(\mathbf{4}\), constant term \(\mathbf{4}\), not monic.
(iii) \(\dfrac{3}{x^2} = 3x^{-2}\). A negative power is not allowed, so this is not a polynomial.
(i) \(P(-1)\)
(ii) \(P(2) - P(-1)\)
(iii) the sum of the coefficients of \(P(x)\).
(i) Replace every \(x\) with \((-1)\), in brackets.
| \(P(-1)\) | \(=\) | \((-1)^3 - 4(-1)^2 + 2(-1) + 5\) |
| \(\) | \(=\) | \(-1 - 4 - 2 + 5\) |
| \(\) | \(=\) | \(\mathbf{-2}\) |
(ii) Find \(P(2)\) first.
| \(P(2)\) | \(=\) | \(8 - 16 + 4 + 5 = 1\) |
| \(P(2) - P(-1)\) | \(=\) | \(1 - (-2)\) |
| \(\) | \(=\) | \(\mathbf{3}\) |
(iii) Every power of \(1\) is \(1\), so \(P(1)\) adds up the coefficients.
| \(P(1)\) | \(=\) | \(1 - 4 + 2 + 5\) |
| \(\) | \(=\) | \(\mathbf{4}\) |
(i) Find \(3A(x) - B(x)\) and state its degree.
(ii) For the product \(A(x)\,B(x)\), state the degree and leading coefficient, and find the coefficient of \(x^2\) without expanding fully.
(i) Multiply every term of \(A(x)\) by \(3\), then subtract every term of \(B(x)\).
| \(3A(x) - B(x)\) | \(=\) | \(3x^3 + 6x^2 - 15\) |
| \(\) | \(\) | \({} - 3x^3 + x - 4\) |
| \(\) | \(=\) | \(\boldsymbol{6x^2 + x - 19}\) |
The \(x^3\) terms cancel, so the degree drops to \(\mathbf{2}\).
(ii) Leading term: \(x^3 \times 3x^3 = 3x^6\). Degree \(\mathbf{6}\), leading coefficient \(\mathbf{3}\).
For \(x^2\), list the pairs (one term from each bracket) whose powers add to \(2\).
| \(2x^2 \times 4\) | \(=\) | \(8x^2\) |
| \(-5 \times (\text{an } x^2 \text{ term})\) | \(\) | \(\text{none: } B \text{ has no } x^2 \text{ term}\) |
The \(x^3\) term of \(A\) can only give powers of \(3\) or more. The coefficient of \(x^2\) is \(\mathbf{8}\).
Each fact gives one equation in \(a\) and \(b\).
| \(P(1) = 1 + a + b + 4\) | \(=\) | \(0\) |
| \(a + b\) | \(=\) | \(-5\) |
| \(P(2) = 8 + 4a + 2b + 4\) | \(=\) | \(6\) |
| \(2a + b\) | \(=\) | \(-3\) |
Subtract the first equation from the second.
| \(a\) | \(=\) | \(\mathbf{2}\) |
| \(b\) | \(=\) | \(-5 - 2 = \mathbf{-7}\) |
Check: \(P(x) = x^3 + 2x^2 - 7x + 4\) gives \(P(1) = 1 + 2 - 7 + 4 = 0\) and \(P(2) = 8 + 8 - 14 + 4 = 6\).
Common pitfalls
Frequently asked questions
What makes something a polynomial?
Every power of the variable must be a whole number: 0, 1, 2 and so on. Negative powers like 1 over x, fractional powers like the square root of x, and x in an index are not allowed. The coefficients can be any real numbers.
What is the degree of a polynomial?
The highest power of \(x\) after the polynomial has been simplified. For example \(2x^5 - x + 1\) has degree \(5\).
What is a monic polynomial?
A polynomial whose leading coefficient is \(1\), such as \(x^3 - 4x + 2\). \(2x^3 + 1\) is not monic.
What does P(2) mean?
It is the value of the polynomial \(P(x)\) when \(x = 2\). Replace every \(x\) with \(2\) and evaluate.
How do you find the degree of a product of polynomials?
Add the degrees. A cubic times a quadratic has degree 5, because the leading terms multiply to give x to the 5.