Monomials, polynomials and special products

A farmer in Kikwit enlarges a square field of side $x$ metres by lengthening each of its sides by $3$ m. The area of the new field is $(x+3)^2$. How many square metres has he gained? To answer, you need to expand such an expression. In this lesson you add, subtract and multiply polynomials, use special products to expand and factorise, factorise a quadratic trinomial and solve a product equation.

Monomials and polynomials

A monomial is the product of a number (the coefficient) and powers of letters (the literal part): in $-5x^3$, the coefficient is $-5$ and the degree is $3$.

Two monomials are like terms if they have the same literal part: $4x^2$ and $-x^2$. Add them by adding the coefficients: $4x^2-x^2=3x^2$.

A polynomial is a sum of monomials. Its degree is the highest degree of its terms: $2x^3-x+7$ has degree $3$.

To subtract a polynomial, change the sign of each of its terms: $-(x^2-4x+1)=-x^2+4x-1$.

Multiplication and special products

• Monomial times monomial: multiply the coefficients and add the exponents: $(3x^2)\times(-2x^3)=-6x^5$.
• Double distributive law: $(a+b)(c+d)=ac+ad+bc+bd$.

Special products:
$$(a+b)^2=a^2+2ab+b^2$$
$$(a-b)^2=a^2-2ab+b^2$$
$$(a+b)(a-b)=a^2-b^2$$

Read from left to right, they are used to expand; read from right to left, they are used to factorise: $x^2-16=(x-4)(x+4)$.

The trinomial $x^2+sx+p$ and product equations

To factorise $x^2+sx+p$, look for two numbers $m$ and $n$ whose sum is $s$ and whose product is $p$. Then:
$$x^2+sx+p=(x+m)(x+n)$$
Example: for $x^2+5x+6$, we need a sum of $5$ and a product of $6$: these are $2$ and $3$, so $x^2+5x+6=(x+2)(x+3)$.

Product equation: a product is zero if and only if one of its factors is zero:
$$A\times B=0\iff A=0\ \text{or}\ B=0$$