SPI Mathematics · Free notes

Polynomials: SPI Mathematics

Polynomials has 32 questions in the SPI bank, all from the SPI sample and practice papers and the LWS mock tests, and none from the 2022 paper. Seventeen are EASY and four are HARD. Very few need long division: most fall to putting one number into the polynomial, or to reading the sum and the product of the zeroes from the coefficients. Learn those two shortcuts well, and keep long division for the questions that ask for a quotient.

The handout has every subtopic, worked example, formula and trap with answers shown. The formula sheet is the chapter’s formulas and traps alone, on one page.

SPI home All formulas on one page

Subtopic notes

Formula & revision sheet

9 formulas · 2 reference tables · 18 gotchas across all subtopics — the exam-eve cheat-sheet

Degree, Value and Zeroes of a Polynomial

Formulas (1)

Value of a polynomial and what a zero is · Zero of a polynomial $p(a) = 0 \qquad \text{zero of } ax + b = -\frac{b}{a}$

Reference tables (2)

Degree of a polynomial and the name for each degree5 rows

Name	Degree	Example
Constant polynomial	0	7
Linear polynomial	1	3x + 2
Quadratic polynomial	2	x² − 4x + 1
Cubic polynomial	3	2x³ + x − 5
Biquadratic (quartic) polynomial	4	x⁴ − 5x² + 6 A biquadratic is a quadratic in x², so its degree is 4, not 2.

The same names are used for equations: a cubic equation has degree 3.

Reading the zeroes of a polynomial from its graph5 rows

What the graph does	Number of distinct zeroes
A straight line crosses the x-axis once	1
A parabola cuts the x-axis at two points	2
A parabola only touches the x-axis at one point	1 Touching counts: the point where it touches is a zero.
A parabola never meets the x-axis	0
A cubic curve crosses the x-axis three times	3

Count the meeting points with the x-axis, not with the y-axis.

Watch out for (5)

Taking the degree from the first term
→ Degree of a polynomial and the name for each degree
Treating a root sign as a power of x
→ Degree of a polynomial and the name for each degree
Squaring after attaching the minus sign
→ Value of a polynomial and what a zero is
Counting where the graph meets the y-axis
→ Reading the zeroes of a polynomial from its graph
Missing a zero where the curve only touches
→ Reading the zeroes of a polynomial from its graph

Factorising Polynomials

Formulas (2)

Factorising a quadratic polynomial by splitting the middle term · Product of two brackets $(x + p)(x + q) = x^2 + (p + q)x + pq$
Factorising by grouping and by substitution · Grouping pattern $ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)$

Watch out for (4)

Checking only the product
→ Factorising a quadratic polynomial by splitting the middle term
Comparing terms of different powers
→ Finding a coefficient by expanding and comparing
Taking out a positive factor from a negative pair
→ Factorising by grouping and by substitution
Leaving the answer in t
→ Factorising by grouping and by substitution

Division Algorithm, Remainder and Factor Theorems

Formulas (3)

The division algorithm for polynomials · Division algorithm $p(x) = g(x)\,q(x) + r(x), \quad \deg r < \deg g \qquad \deg q = \deg p - \deg g$
The remainder theorem · Remainder theorem $\text{remainder of } p(x) \div (x - a) = p(a)$
The factor theorem · Factor theorem $(x - a) \text{ is a factor of } p(x) \text{ when } p(a) = 0$

Watch out for (4)

Skipping a missing power
→ The division algorithm for polynomials
Expecting a number as every remainder
→ The division algorithm for polynomials
Putting in the wrong sign
→ The remainder theorem
Mixing up the factor and the zero
→ The factor theorem

Zeroes and Coefficients

Formulas (3)

Sum and product of the zeroes of a quadratic polynomial · Zeroes of a quadratic polynomial $\alpha + \beta = -\frac{b}{a} \qquad \alpha\beta = \frac{c}{a} \qquad \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$
Forming a quadratic polynomial from its zeroes · Quadratic from its zeroes $k\left[x^2 - (\alpha + \beta)x + \alpha\beta\right]$
Zeroes and coefficients of a cubic polynomial · Zeroes of a cubic polynomial $\alpha + \beta + \gamma = -\frac{b}{a} \qquad \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \qquad \alpha\beta\gamma = -\frac{d}{a}$

Watch out for (5)

Dropping the minus in the sum
→ Sum and product of the zeroes of a quadratic polynomial
Reading signs from the product alone
→ Sum and product of the zeroes of a quadratic polynomial
Thinking only one polynomial fits
→ Forming a quadratic polynomial from its zeroes
Adding instead of subtracting in a conjugate product
→ Forming a quadratic polynomial from its zeroes
Using the quadratic sign for a cubic product
→ Zeroes and coefficients of a cubic polynomial

Question weightage by concept

12 concepts · 32 questions — where the marks actually sit, so you know what to drill first

Degree, Value and Zeroes of a Polynomial8 questions · 25%

Concept	Qs	Share
Degree of a polynomial and the name for each degree	4	13%
Value of a polynomial and what a zero is	2	6%
Reading the zeroes of a polynomial from its graph	2	6%

Factorising Polynomials5 questions · 16%

Concept	Qs	Share
Factorising a quadratic polynomial by splitting the middle term	2	6%
Factorising by grouping and by substitution	2	6%
Finding a coefficient by expanding and comparing	1	3%

Division Algorithm, Remainder and Factor Theorems10 questions · 31%

Concept	Qs	Share
The division algorithm for polynomials	6	19%
The factor theorem	3	9%
The remainder theorem	1	3%

Zeroes and Coefficients9 questions · 28%

Concept	Qs	Share
Forming a quadratic polynomial from its zeroes	4	13%
Zeroes and coefficients of a cubic polynomial	3	9%
Sum and product of the zeroes of a quadratic polynomial	2	6%

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