Factor polynomial $$ 146a^{3}-1457a^{2}+4464a-3888 $$
Answer
Explanation
Step 1 :
The polynomial $ 146a^{3}-1457a^{2}+4464a-3888 $ can be factorized by using the rational root test. To apply this test first we need to find
at least one rational zero.
Polynomial roots calculator can be used to find rational zero.
In this case one rational zero is $ a = 4 $.
This means that we can divide polynomial $ 146a^{3}-1457a^{2}+4464a-3888 $ by $ a-4 $ (the factor theorem)
After division we have: (you can use Synthetic Division Calculator for the step-by-step explanation on how to divide polynomials. )
$$ \frac{ 146a^{3}-1457a^{2}+4464a-3888 }{ a-4 } = 146a^{2}-873a+972 $$that is:
$$ 146a^{3}-1457a^{2}+4464a-3888 = ( a-4 )( 146a^{2}-873a+972 ) $$Step 2 :
Step 2: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:
Step 3: Multiply the leading coefficient $\color{blue}{ a = 146 }$ by the constant term $\color{blue}{c = 972} $.
$$ a \cdot c = 141912 $$Step 4: Find out two numbers that multiply to $ a \cdot c = 141912 $ and add to $ b = -873 $.
Step 5: All pairs of numbers with a product of $ 141912 $ are:
| PRODUCT = 141912 | |
| 1 141912 | -1 -141912 |
| 2 70956 | -2 -70956 |
| 3 47304 | -3 -47304 |
| 4 35478 | -4 -35478 |
| 6 23652 | -6 -23652 |
| 8 17739 | -8 -17739 |
| 9 15768 | -9 -15768 |
| 12 11826 | -12 -11826 |
| 18 7884 | -18 -7884 |
| 24 5913 | -24 -5913 |
| 27 5256 | -27 -5256 |
| 36 3942 | -36 -3942 |
| 54 2628 | -54 -2628 |
| 72 1971 | -72 -1971 |
| 73 1944 | -73 -1944 |
| 81 1752 | -81 -1752 |
| 108 1314 | -108 -1314 |
| 146 972 | -146 -972 |
| 162 876 | -162 -876 |
| 216 657 | -216 -657 |
| 219 648 | -219 -648 |
| 243 584 | -243 -584 |
| 292 486 | -292 -486 |
| 324 438 | -324 -438 |
Step 6: Find out which factor pair sums up to $\color{blue}{ b = -873 }$
| PRODUCT = 141912 and SUM = -873 | |
| 1 141912 | -1 -141912 |
| 2 70956 | -2 -70956 |
| 3 47304 | -3 -47304 |
| 4 35478 | -4 -35478 |
| 6 23652 | -6 -23652 |
| 8 17739 | -8 -17739 |
| 9 15768 | -9 -15768 |
| 12 11826 | -12 -11826 |
| 18 7884 | -18 -7884 |
| 24 5913 | -24 -5913 |
| 27 5256 | -27 -5256 |
| 36 3942 | -36 -3942 |
| 54 2628 | -54 -2628 |
| 72 1971 | -72 -1971 |
| 73 1944 | -73 -1944 |
| 81 1752 | -81 -1752 |
| 108 1314 | -108 -1314 |
| 146 972 | -146 -972 |
| 162 876 | -162 -876 |
| 216 657 | -216 -657 |
| 219 648 | -219 -648 |
| 243 584 | -243 -584 |
| 292 486 | -292 -486 |
| 324 438 | -324 -438 |
Step 7: Replace middle term $ -873 x $ with $ -216x-657x $:
$$ 146x^{2}-873x+972 = 146x^{2}-216x-657x+972 $$Step 8: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -9 $ out of the last two terms.
$$ 146x^{2}-216x-657x+972 = 2x\left(73x-108\right) -9\left(73x-108\right) = \left(2x-9\right) \left(73x-108\right) $$