Special products of polynomials

In the previous section we showed you how to multiply binominals. There are a couple of special instances where there are easier ways to find the product of two binominals than multiplying each term in the first binomial with all terms in the second binomial.

Look what happens when you square a binomial.

$$\left ( x+2 \right )^{2}=$$

$$=\left ( x+2 \right )\left ( x+2 \right )=$$

$$=x^{2}+2x+2x+4=$$

$$=x^{2}+4x+4$$

$$=x^{2}+\left ( 2\cdot 2\cdot x \right )+2^{2}$$

This is a pattern that's called the square of a binomial pattern.

$$\left ( x{\color{green} \, +\, }y \right )^{2}=x^{2}{\color{green} {\, +\, }2xy+y^{2}}$$

$$\left ( x{\color{green} \, -\, }y \right )=x^{2}{\color{green} {\, -\, }2xy+y^{2}}$$

There is another pattern that is good to know. We begin by looking at an example. What happens if we multiply two binominals where one is a sum of two terms and the other is the different between the same two terms?

$$\left ( x+5 \right )\left ( x-5 \right )=x^{2}-{\color{red} {\not}{5x}}+{\color{red} {\not}{5x}-25=x^{2}-25=x^{2}-5^{2}}$$

This is called the sum and difference pattern.

$$\left ( x+y \right )\left ( x-y \right )=x^{2}-y^{2}$$


Video lesson

Evolve the following expression

$$2(x+3)^{2}$$