Form 3

Basic Concept

The Square of The Sum of Polynomials

$$(a + b + c\dotso)^2 = \text{square of each elements + two times of each two elements}$$

Proof:

\begin{equation*}
\begin{split}
(a + b + c)^2 & = (a + b + c)(a + b + c)\\\\
(a + b + c)^2& = a(a + b + c) + b(a + b + c) + c(a + b + c)\\\\
(a + b + c)^2& = a^2 + ab + ac + ab + b^2 + bc + ac + bc + c^2\\\\
(a + b + c)^2& = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc
\end{split}
\end{equation*}

Example 1

Expand $$(2a + 3b + c)^2$$

\begin{equation*}
\begin{split}
&(2a)^2 + (3b)^2 + c^2 + 2(2a)(3b) + 2(2a)(c) + 2(3b)(c)\\\\
&4a^2 + 9b^2 + c^2 + 12ab + 4ac + 6bc
\end{split}
\end{equation*}

Example 2

Expand $$(m - 2n + 5r)^2$$

\begin{equation*}
\begin{split}
& (m + (-2n) + 5r)^2\\\\
&m^2 + (-2n)^2 + (5r)^2 + 2(m)(-2n) + 2(m)(5r) + 2(-2n)(5r)\\\\
& m^2 + 4n^2 + 25r^2 - 4mn + 10mr - 20nr
\end{split}
\end{equation*}

Factorize $$a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$$

$$\color{blue}1^{st}$$ step:  root each element which has power of two

$$a^2 \rightarrow \sqrt{a^2} = (a) \text{ or } (-a)$$

$$b^2 \rightarrow \sqrt{b^2} = (b) \text{ or } (-b)$$

$$c^2 \rightarrow \sqrt{c^2} = (c) \text{ or } (-c)$$

$$\color{blue}2^{nd}$$ step:  look at the two times of each two elements

$$2ab \rightarrow 2(a)(b) \text{ or } 2(-a)(-b)$$

$$2ac \rightarrow 2(a)(c) \text{ or } 2(-a)(-c)$$

$$2bc \rightarrow 2(b)(c) \text{ or } 2(-b)(-c)$$

$$\color{blue}3^{rd}$$ step:  There are two option for the answer

$$\color{purple} 1^{st}\text{ option}$$

$$\text{The positive element } a^2 + b^2 + c^2 + 2ab + 2ac + 2bc = (a + b + c)^2$$

$$\color{purple} 2^{nd}\text{ option}$$

$$\text{The negatif element } a^2 + b^2 + c^2 + 2ab + 2ac + 2bc = (-a - b - c)^2$$

Example

Factorize $$x^2 + y^2 + 16 + 2xy + 8x + 8y$$

$$\color{blue}1^{st}$$ step:  root each element which has power of two

$$x^2 \rightarrow \sqrt{x^2} = x$$

$$y^2 \rightarrow \sqrt{y^2} = y$$

$$16 \rightarrow \sqrt{4^2} = 4$$

$$\color{blue}2^{nd}$$ step:  look at the two times of each two elements

$$2xy \rightarrow 2(x)(y)$$

$$8x \rightarrow 2(x)(4)$$

$$8y \rightarrow 2(4)(y)$$

$$\color{blue}3^{rd}$$ step:  $$x^2 + y^2 + 16 + 2xy + 8x + 8y = (x + y + 4)^2$$