Degree and value
Before any operation, write the polynomial in descending powers of x and name its degree, leading coefficient and constant term. To work out P(a), substitute x, with brackets when a is negative.
Start over?
Teaching Guide: Polynomials from Grade 8 to Grade 10
Polynomials are the language of secondary school algebra: expressions are added, multiplied and divided as if they were numbers, and factorising them paves the way for quadratic equations and algebraic fractions. This free polynomials worksheet generator makes A4 worksheets on degree and value, the four operations, long division, synthetic division, the remainder and factor theorems, special products both ways and factorising, for Grades 8 to 10, with a fully worked answer key.
Student worksheet 
Answer key Add, subtract and multiply
Line up like terms in columns; to subtract, add the opposite. To multiply, take each term times the whole bracket and collect like terms, or use the column method as with numbers.
Long division
Divide the first term, multiply the divisor by it, subtract and repeat until the remainder has a lower degree than the divisor. The answer key shows the whole bus stop layout.
Synthetic division
To divide by x − a, the coefficients are enough: synthetic division (Ruffini's rule) gives the quotient and the remainder, which is P(a).
Special products
(a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b² and (a + b)(a − b) = a² − b², to expand and, above all, to spot inside a polynomial.
Factorising
First the common factor, then special products and, when needed, synthetic division with the factors of the constant term until every factor is linear.
Frequently Asked Questions about the Polynomials Worksheets
Which grades are these worksheets for?
Grades 8 to 10. Grade 8 covers degree, value, adding and subtracting, a monomial times a polynomial and special products with x; Grade 9 products of polynomials, long division and synthetic division; Grade 10 division by any polynomial, the remainder and factor theorems and factorising with integer roots.
Do the divisions work out exactly?
The coefficients are always whole numbers: each dividend is built as divisor times quotient plus remainder, so long division and synthetic division never leave the integers, even when the divisor's leading coefficient is not 1.
How is long division set out in the answer key?
In the British bus stop layout: the divisor outside the bracket, the quotient on top with each term above the term of the same degree, and each product written underneath and subtracted until the remainder has a lower degree than the divisor.
What does the synthetic division table show?
The usual three-row table with a 0 for each missing term, the value of a on the left and the remainder highlighted. At level 3 the remainder is checked with the remainder theorem by working out P(a); when factorising, the tables are chained, one for each root.
What is the difference between expanding and writing as a special product?
Expanding takes (2x − 3)² to 4x² − 12x + 9. Going back the other way, spotting that 4x² − 12x + 9 is a perfect square, is much harder and is the key to factorising and simplifying algebraic fractions, so it has an exercise type of its own.
Which other worksheets go well with these?
Powers and roots revise the laws of indices, integer operations the signs, and next come quadratic equations, solved by factorising, and simultaneous equations. Find every resource in maths worksheets.
Is any student data sent anywhere?
No. The worksheets are made in your browser and templates are saved in your browser's storage.
More worksheet generators you may like
Linear Equations
One-step and two-step linear equations, x on both sides, brackets and fractions, with whole, negative or fraction solutions and a step-by-step answer key with the check.
Quadratic Equations
Incomplete quadratics and the quadratic formula, equations with brackets and fractions to rearrange, the number of solutions from the discriminant and the equation from its roots, with a step-by-step answer key.
Systems of Equations
Systems of two equations in two unknowns (simultaneous equations) by substitution, equalisation and elimination, the graphical method, classifying systems and word problems, with a step-by-step answer key.