factorise 2x^3 - x^2 - 16x +5 using factor theoram
Answers
Step-by-step explanation:
To factorise
\begin{gathered}2 {x}^{3} - {x}^{2} - 16x + 15 \\ \\\end{gathered}
2x
3
−x
2
−16x+15
The first factor can be find using hit and trial method; for x= 1
\begin{gathered}= 2 {(1)}^{3} - {(1)}^{2} - 16(1) + 15 \\ \\ = 2 - 1 - 16 + 15 \\ \\ = 17 - 17 \\ \\ = 0 \\ \\\end{gathered}
=2(1)
3
−(1)
2
−16(1)+15
=2−1−16+15
=17−17
=0
hence (x-1) is a factor of polynomial.
Now as we know that it's a polynomial of degree three,so it should have three factors.
To find the other to factors
\begin{gathered}x - 1 \: )2 {x}^{3} - {x}^{2} - 16x + 15(2 {x}^{2} + x - 15 \\ \: \: \: \: \: \: \: \: \: \: \: \: 2 {x}^{3} - 2 {x}^{2} \\ \: \: \: \: \: \: \: \: ( -) \: \: \: \: ( + ) \\ \: \: \: \: \: \: \: \: - - - - - \\ \: \: \: \: \: \: \: \: {x}^{2} - 16x \\ \: \: \: \: \: \: \: \: {x}^{2} - x \\ \: \: \: \: \: ( - ) \: \: ( + ) \\ \: \: \: \: \: \: \: - - - - - \\ \: \: \: \: \: \: \: \: \: \: - 15x + 15 \\ \: \: \: \: \: \: \: \: \: - 15x + 15 \\ \: \: \: \: \: \: \: ( + ) \: \: \: ( - ) \\ \: \: \: \: \: \: - - - - - - \\ \: \: \: \: \: \: \: \: \: \: \: 0 \\ \\\end{gathered}
x−1)2x
3
−x
2
−16x+15(2x
2
+x−15
2x
3
−2x
2
(−)(+)
−−−−−
x
2
−16x
x
2
−x
(−)(+)
−−−−−
−15x+15
−15x+15
(+)(−)
−−−−−−
0
So now factorise
\begin{gathered}2 {x}^{2} + x - 15 \\ \\ 2 {x}^{2} + 6x - 5x - 15 \\ \\ 2x(x + 3) - 5(x + 3) \\ \\ (x + 3)(2x - 5) \\ \\\end{gathered}
2x
2
+x−15
2x
2
+6x−5x−15
2x(x+3)−5(x+3)
(x+3)(2x−5)
Thus all three factors of given polynomial are
\begin{gathered}(x + 3)(2x - 5)(x - 1) \\ \\\end{gathered}
(x+3)(2x−5)(x−1)
Hope it helps you.