Mathematics · Expressions, equations and inequalities · Grade 7

Monomials and polynomials

The vegetable plot was extended by 5 m: which pieces make up the area (x + 3)(x + 5)?

The situation

The vegetable plot was x metres by (x + 2) metres. We added 3 metres from the neighbouring land, making the length x + 3 metres. The new area is (x + 3)(x + 2). If we draw it, we get four pieces: x², 3x, 2x and 6 — each one a monomial, and their sum a polynomial. If we put the pieces 3x and 2x side by side, they make 5x. How do we work with polynomials: how can we add, multiply, divide them and break them into pieces again?

Topic

Monomials and polynomials: standard form, adding, subtracting, multiplying, dividing, factorising. The area of an extended plot of land: the area model of (x + 3)(x + 5) — the pieces of the rectangle become the terms of the polynomial

Key ideas

Standard form
Area model
Factorising

Formulas

x^{m} \cdot x^{n} = x^{m + n}, 2 x \cdot 3 x = 6 x^{2}

Multiplying monomials: the coefficients are multiplied and the powers of the same letters are added.

(a + b) (c + d) = a c + a d + b c + b d

Multiplying polynomials — the area model: every term is multiplied by every term.

(x + 3) (x + 2) = x^{2} + 3 x + 2 x + 6 = x^{2} + 5 x + 6

Like terms ( $3 x$ and $2 x$ ) combine — the standard form of a polynomial.

6 x^{2} + 9 x = 3 x (2 x + 3), (x^{2} + 5 x + 6) : (x + 2) = x + 3

Factorising and dividing are the inverse of multiplying: from the area back to the sides.

Theory

The monomial and its standard form

A product of numbers and letters is a monomial: $3 x^{2} y$ , $- 5 a$ , $7$ . Standard form: the coefficient first, each letter once, with its power: $x \cdot 3 \cdot y \cdot x = 3 x^{2} y$ . The numerical factor is the coefficient (3), the letters are the letter part. The degree of a monomial is the sum of the powers of its letters: for $3 x^{2} y$ it is $2 + 1 = 3$ .

Multiplying and dividing monomials

$2 x \cdot 3 x$ is a rectangle with sides $2 x$ and $3 x$ : 6 squares of $x^{2}$ , so $6 x^{2}$ . The rule: the coefficients are multiplied and the powers of the same letters are added: $x^{m} \cdot x^{n} = x^{m + n}$ . Division is the inverse: the powers are subtracted, $6 x^{2} : 2 x = 3 x$ (area $6 x^{2}$ , length $2 x$ — width $3 x$ ); the divisor must not be zero, that is, $x \neq 0$ . For the result to be a monomial, the coefficients must divide exactly and the powers in the dividend must not be smaller than those in the divisor.

Polynomial, like terms, standard form

A sum of monomials is a polynomial: $x^{2} + 3 x + 2 x + 6$ . Terms with the same letter part are like terms ( $3 x$ and $2 x$ ); combining them is simplifying: $5 x$ . A polynomial with no like terms and every term in standard form is a polynomial in standard form; the terms are usually written in decreasing order of degree: $x^{2} + 5 x + 6$ . The degree of a polynomial is the greatest degree of its terms (2).

Adding, subtracting, multiplying

When adding, the brackets are simply removed; when subtracting, every term of the polynomial being subtracted changes sign: $(2 x^{2} + 3 x - 1) - (x^{2} - 5 x + 4) = 2 x^{2} + 3 x - 1 - x^{2} + 5 x - 4 = x^{2} + 8 x - 5$ . Multiplying — the area model: the rectangle $(x + 3) (x + 5)$ consists of the pieces $x \cdot x$ , $x \cdot 5$ , $3 \cdot x$ , $3 \cdot 5$ — every term is multiplied by every term, then the like terms are collected.

Dividing and factorising

To divide a polynomial by a monomial, divide each term separately: $(6 x^{2} + 4 x) : 2 x = 3 x + 2$ . To divide a polynomial by a polynomial is the inverse problem in the area model: if we fit the pieces into a rectangle of length $x + 2$ , the width $x + 3$ is the quotient. Factorising means writing a polynomial as a product: taking out the common factor ( $6 x^{2} + 9 x = 3 x (2 x + 3)$ ), grouping ( $a b + a c + b^{2} + b c = (a + b) (b + c)$ ). To check, multiply out again.

About the model

In the drawings $x$ is a definite length (in fact any number), and negative terms are shown conventionally by red tiles: there is no negative area, it is only a sign marker. The coefficients on the sliders are small whole numbers; division is shown only when there is no remainder (division with a remainder comes in later grades). The special products formulae have an experiment of their own.

Did you know?

The Uzbek term for a polynomial literally means “many terms”; it translates the word “polynomial”: Greek “poly” (many) and Latin “nomen” (name, term). A polynomial with two terms is a binomial, with three a trinomial.
Long multiplication of numbers is really multiplication of polynomials: $23 \cdot 45 = (2 \cdot 10 + 3) (4 \cdot 10 + 5)$ , that is, the value of $(2 x + 3) (4 x + 5)$ at $x = 10$ .

Inside the experiment

The topic is explained step by step with animation, from simple to advanced: from a real-life picture to the formula and the proof. Then you drag the points yourself to check. Inside are missions, a quiz and a guided tour.

More experiments in this section

Expressions: from numbers to letters — Grade 7
Special products (algebraic identities) — Grade 7
Algebraic fractions — Grade 7
The linear equation: the balance — Grade 7
Systems of equations — Grade 7
Inequalities: ray and interval — Grade 8
The quadratic equation: the parabola — Grade 8