x⁴-7x³+9x²+7x-10 given that (x-1) is a factor
Answers
The factorised polynomial to the given polynomial is x^4-7x^3+9x^2+7x-10=(x-1)(x+1)(x-2)(x-5)x
4
−7x
3
+9x
2
+7x−10=(x−1)(x+1)(x−2)(x−5)
Step-by-step explanation:
Given polynomial is x^4-7x^3+9x^2+7x-10x
4
−7x
3
+9x
2
+7x−10
To factorise the given polynomial by Factor method :
Let f(x) be the given polynomial
Therefore f(x)=x^4-7x^3+9x^2+7x-10f(x)=x
4
−7x
3
+9x
2
+7x−10
Since the degree of the given polynomial is 4
Therefore it has 4 factors
Put x=1 in the above polynomial we get
f(1)=1^4-7(1)^3+9(1)^2+7(1)-10f(1)=1
4
−7(1)
3
+9(1)
2
+7(1)−10
=1-7+9+7-10=1−7+9+7−10
Therefore f(1)=0
Therefore x-1 is a factor
By factor theorem " If f(x) is a polynomial and f(1)=0 then x-1 is a factor of f(x) "
x^3-6x^2+3x+10x
3
−6x
2
+3x+10
____________________
x-1) x^4-7x^3+9x^2+7x-10x
4
−7x
3
+9x
2
+7x−10
x^4-x^3x
4
−x
3
(-)__(+)______________
-6x^3+9x^2−6x
3
+9x
2
-6x^3+6x^2−6x
3
+6x
2
_____ (+)__(-)___________
3x^2+7x3x
2
+7x
3x^2-3x3x
2
−3x
__________(-)__(+)___________
10x-1010x−10
10x-1010x−10
_(-)__(+)_________
0
____________________
Therefore we have x^3-6x^2+3x+10x
3
−6x
2
+3x+10
Put x=-1 in the above polynomial we get
f(-1)=(-1)^4-7(-1)^3+9(-1)^2+7(-1)-10f(−1)=(−1)
4
−7(−1)
3
+9(−1)
2
+7(−1)−10
=1+7+9-7-10=1+7+9−7−10
Therefore f(-1)=0
Therefore x+1 is also a factor
By Factor theorem we have that
x^2-7x+10x
2
−7x+10
____________________
x+1) x^3-6x^2+3x+10x
3
−6x
2
+3x+10
x^3+x^2x
3
+x
2
_(-)_(-)______________
-7x^2+3x−7x
2
+3x
-7x^2-7x−7x
2
−7x
_____(+)__(+)___________
10x+1010x+10
10x+1010x+10
_(-)__(-)_________
0
_________________
Therefore we have the quadratic equation x^2-7x+10=0x
2
−7x+10=0
x^2-7x+10=0x
2
−7x+10=0
(x-2)(x-5)=0(x−2)(x−5)=0
Therefore x-2 and x-5 are also factors of f(x)
By using the factor theorem we have factorised the given polynomial as (x-1)(x+1)(x-2)(x-5)
The factorised polynomial to the given polynomial is x^4-7x^3+9x^2+7x-10=(x-1)(x+1)(x-2)(x-5)x
4
−7x
3
+9x
2
+7x−10=(x−1)(x+1)(x−2)(x−5)
Step-by-step explanation:
The factorised polynomial to the given polynomial is
Step-by-step explanation:
Given polynomial is
To factorise the given polynomial by Factor method :
Let f(x) be the given polynomial
Therefore
Since the degree of the given polynomial is 4
Therefore it has 4 factors
Put x=1 in the above polynomial we get
Therefore f(1)=0
Therefore x-1 is a factor
By factor theorem " If f(x) is a polynomial and f(1)=0 then x-1 is a factor of f(x) "
____________________
x-1)
(-)__(+)______________
_____ (+)__(-)___________
__________(-)__(+)___________
_(-)__(+)_________
0
____________________
Therefore we have
Put x=-1 in the above polynomial we get
Therefore f(-1)=0
Therefore x+1 is also a factor
By Factor theorem we have that
____________________
x+1)
_(-)_(-)______________
_____(+)__(+)___________
_(-)__(-)_________
0
_________________
Therefore we have the quadratic equation
Therefore x-2 and x-5 are also factors of f(x)
By using the factor theorem we have factorised the given polynomial as (x-1)(x+1)(x-2)(x-5)
The factorised polynomial to the given polynomial is