# How do you factor 6r^2-28r+16?

Jun 2, 2015

$6 {r}^{2} - 28 r + 16$

$= 2 \left(3 {r}^{2} - 14 r + 8\right)$

To factor $3 {r}^{2} - 14 r + 8$ use a modified AC Method...

$A = 3$, $B = 14$, $C = 8$

Look for a factorization of $A C = 3 \times 8 = 24$ into a pair of factors whose sum is $B = 14$.

The pair $B 1 = 2$, $B 2 = 12$ works.

Then for each of the combinations: $\left(A , B 1\right)$ and $\left(A , B 2\right)$, divide by the HCF (highest common factor) to get the coefficients of a factor of $3 {r}^{2} - 14 r + 8$ ...

$\left(3 , 2\right)$ (HCF $1$) $\to \left(3 , 2\right) \to \left(3 r - 2\right)$
$\left(3 , 12\right)$ (HCF $3$) $\to \left(1 , 4\right) \to \left(r - 4\right)$

So

$6 {r}^{2} - 28 r + 16 = 2 \left(3 r - 2\right) \left(r - 4\right)$

Aug 26, 2016

$2 \left(3 x - 2\right) \left(x - 4\right)$

#### Explanation:

Factoring requires a solid knowledge of the multiplication tables.

Always look for a common factor first - all these numbers are even.

$6 {r}^{2} - 28 r + 16 = 2 \left(3 {r}^{2} - 14 r + 8\right) \text{ factor the trinomial}$

All the clues are in the trinomial o3 3 and 8. Compare the colors and read it as follows:

$\textcolor{\mathmr{and} a n \ge}{3} \textcolor{b l u e}{-} \textcolor{red}{14} \textcolor{\lim e}{+} \textcolor{\mathmr{and} a n \ge}{8}$

Find the factors of $\textcolor{\mathmr{and} a n \ge}{3 \mathmr{and} 8}$ which $\textcolor{\lim e}{\text{ADD}}$ to give $\textcolor{red}{14}$

The signs will be $\textcolor{\lim e}{\text{THE SAME}}$, they are both$\textcolor{b l u e}{\text{ minus}}$

Use different factors of 3 and 8 and cross-multiply with different combinations, until the sum of the products is 14.

$\textcolor{w h i t e}{\times \times} \textcolor{\mathmr{and} a n \ge}{\left(3\right) \text{ } \left(8\right)}$
$\textcolor{w h i t e}{\times \times x} 3 \text{ } 2 \rightarrow 1 \times 2 = 2$
$\textcolor{w h i t e}{\times \times x} 1 \text{ } 4 \rightarrow 3 \times 4 = \underline{12}$
$\textcolor{w h i t e}{\times \times \times \times \times \times \times \times \times} 14 \leftarrow \text{we have the correct factors}$

Now put the signs into the brackets.

$2 \left(3 x - 2\right) \left(x - 4\right)$