Expressions that mix in letters such as x and y follow the same rules as arithmetic, so a handful of rules is enough to shrink a long expression into a short one.
An expression made of a single product of numbers and letters, such as 3x², is a monomial, and a sum of monomials is a polynomial.
Terms with exactly the same letter part, called like terms, can be gathered like blocks of the same shape so that only their coefficients add, and multiplying a polynomial by a monomial means multiplying term by term with the distributive law, the way the area of a rectangle splits into pieces.
The blocks and the area model below show why these two rules hold.
Products of numbers and letters
👀 Monomials and polynomials
①An expression that is only a product of numbers and letters is a monomial: 3x², −5xy and 7 are all monomials.
②In 3x², 3 is the coefficient, x² is the letter part, and the number of letters multiplied, 2, is the degree.
③An expression such as 3x² + 2x − 1, a sum of several monomials, is a polynomial, and each monomial added is a term.
Collecting like terms
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Lining up identical x blocks so that their counts, the coefficients, add
3
2
Adding like terms
axn + bxn = (a + b)xn
Only terms whose letters and degrees are all the same can have their coefficients added. 3x + 2x = 5x, but 3x + 2 cannot be simplified further.
Polynomial × monomial as area
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A rectangle of height x and width ax + b divided into x × x squares and 1 × x strips
3
2
Distributive law
a(b + c) = ab + ac
Multiply the term outside the brackets by every term inside, one at a time. In the picture, the area of the large rectangle equals the area of the squares plus the area of the strips.
Multiplying and dividing monomials
Multiplying monomials
(multiply the coefficients) × (index laws for each letter)
In 3x² × 2x³, the coefficients give 3 × 2 = 6 and the indices give 2 + 3 = 5, so the product is 6x⁵.
Dividing monomials
6x52x2 = 3x3
Divide the coefficients and subtract the indices of each letter. You can also turn the division into multiplication by the reciprocal.
Worked examples
Example 1
Compute 3x2 × 4x3.
1
Group coefficients with coefficients and letters with letters, then multiply.
= (3 × 4) × (x2 × x3)
2
Add the indices of the same letter x.
= 12x5
▸ 12x5
A product of monomials is calculated by collecting coefficients together and the same letters together.
Example 2
Expand 2x(3x − 5).
1
Use the distributive law to multiply each of the two terms in the brackets by 2x.
= 2x × 3x − 2x × 5
2
Simplify each product.
= 6x2 − 10x
▸ 6x2 − 10x
(monomial) × (polynomial) is expanded with the distributive law, just like cutting the rectangle in the area model into two pieces.
Exam checklist
Three rules behind every calculation
collect like terms + distributive law + index laws
Most algebraic calculations are just these three applied in order.
Grade-8 school exam type
Compute 12a5 b3 ÷ 3a2 b.
①4a3 b2
②4a3 b3
③9a3 b2
④4a7 b4
⑤36a7 b4
▸ ① 4a3 b2
1
Divide the coefficients and subtract the indices of each letter.
= (12 ÷ 3) × a5−2 × b3−1
2
Work out each part and simplify.
= 4a3 b2
🎯 Exam points
①Like terms must have exactly the same letters and degrees; x² and x are not like terms.
②When using the distributive law, multiply the sign in front of the brackets too.
③When multiplying monomials, multiply the coefficients and add the indices of the same letter.
④When dividing monomials, divide the coefficients and subtract the indices of the same letter.
⑤Sign slips are the most common error: −2(3x − 4) = −6x + 8, so the sign of the second term changes as well.