B. Factoring Trinomials: Factoring by Inspection
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B. Factoring Trinomials: Factoring by Inspection
This method is used mainly to factor trinomials of the form \(ax^2 + bx + c \), where \(a = 1 \).
Factoring and expanding are reverse processes, so start by imagining the multiplication of two factors, and then look to see how you could work backwards from the product.
Suppose a trinomial could be factored into \((x + r) \) and \((x + s) \).
Multiplying these factors gives
\(\begin{align}
\left( {x + r} \right)\left( {x + s} \right) &= x^2 + sx + rx + rs \\
&= x^2 + \left( {{\color{blue}r + s}} \right)x + \color{red}rs \\
\end{align}\)
Notice that \(\color{blue}r + s \) is the \(b\)-value and \(\color{red}rs \) is the \(c\)-value of the trinomial \(x^2 + bx + c \). This means that if there are two numbers, \(r \) and \(s, \) that add to give \(b\) and multiply to give \(c\), then the trinomial \(x^2 + bx + c \) can be factored as \((x + r)(x + s) \).
Factoring and expanding are reverse processes, so start by imagining the multiplication of two factors, and then look to see how you could work backwards from the product.
Suppose a trinomial could be factored into \((x + r) \) and \((x + s) \).
Multiplying these factors gives
\(\begin{align}
\left( {x + r} \right)\left( {x + s} \right) &= x^2 + sx + rx + rs \\
&= x^2 + \left( {{\color{blue}r + s}} \right)x + \color{red}rs \\
\end{align}\)
Notice that \(\color{blue}r + s \) is the \(b\)-value and \(\color{red}rs \) is the \(c\)-value of the trinomial \(x^2 + bx + c \). This means that if there are two numbers, \(r \) and \(s, \) that add to give \(b\) and multiply to give \(c\), then the trinomial \(x^2 + bx + c \) can be factored as \((x + r)(x + s) \).