What must be added to polynomial, ( ) 4 2 2 2 1 4 3 3 2 f x = x + x − x − x + x − so that the resulting
polynomial is exactly divisible by x2 + 2x – 3
Answers
Answer:t P(x)=X^4+2X^3–2X^2+X-1
G(x)=X^2+2X-3
=>X^2+(3–1)X-3
=>X^2+3X-X-3
=>X(X+3)-1(X+3)
=>(X+3)(X-1)
If ax+b added to P(x), then P(x) must be divisible by G(x)
Therefore(x+3) and(x-1) will be the factor of
X^4+2X^3–2X^2+X-1+aX+b
=>(-3)^4+2(-3)^3–2(-3)^2+(-3)-1+a(-3)+b
=>81–54–18–3–1–3a+b=0.
=>5–3a+b=0…….…..…(1)
(1)^4+2(1)^3–2(1)^2+1–1+a+b=0
1+2–2+1–1+a+b=0
1+a+b=0……………(2)
Subtracting(1) from(2)
-4+4a=0
=>a=1
b=-2
So x-2 must be added.
2.7k views · View 2 Upvoters · Answer requested by Z H Ratlamwala and Ajit John
Anca Horsfall
Anca Horsfall
Answered Apr 9, 2018 · Author has 625 answers and 329.2k answer views
Originally Answered: What must be added to the polynomial x4+2x3-2x2+x-1 so that the resulting polynomial is exactly divisible by x2+2x-3?
What must be added to the polynomial x4+2x3-2x2+x-1 so that the resulting polynomial is exactly divisible by x2+2x-3?
You can either do a long division, and add the negative of the remainder to the original polynomial… or, rearrange the polynomial to feature as a factor x2+2x−3x2+2x−3:
x4+2x3−2x2+x−1=x4+2x3–3x2+x2+2x−3−x+2=x4+2x3−2x2+x−1=x4+2x3–3x2+x2+2x−3−x+2=
=x2(x2+2x−3)+(x2+2x−3)−x+2=(x2+1)(x2+2x−3)−x+2=x2(x2+2x−3)+(x2+2x−3)−x+2=(x2+1)(x2+2x−3)−x+2
So it looks like if you add x−2x−2 to your original polynomial, it becomes a multiple of x2+2x−
Answer:t P(x)=X^4+2X^3–2X^2+X-1
G(x)=X^2+2X-3
=>X^2+(3–1)X-3
=>X^2+3X-X-3
=>X(X+3)-1(X+3)
=>(X+3)(X-1)
If ax+b added to P(x), then P(x) must be divisible by G(x)
Therefore(x+3) and(x-1) will be the factor of
X^4+2X^3–2X^2+X-1+aX+b
=>(-3)^4+2(-3)^3–2(-3)^2+(-3)-1+a(-3)+b
=>81–54–18–3–1–3a+b=0.
=>5–3a+b=0…….…..…(1)
(1)^4+2(1)^3–2(1)^2+1–1+a+b=0
1+2–2+1–1+a+b=0
1+a+b=0……………(2)
Subtracting(1) from(2)
-4+4a=0
=>a=1
b=-2
So x-2 must be added.
2.7k views · View 2 Upvoters · Answer requested by Z H Ratlamwala and Ajit John
Anca Horsfall
Anca Horsfall
Answered Apr 9, 2018 · Author has 625 answers and 329.2k answer views
Originally Answered: What must be added to the polynomial x4+2x3-2x2+x-1 so that the resulting polynomial is exactly divisible by x2+2x-3?
What must be added to the polynomial x4+2x3-2x2+x-1 so that the resulting polynomial is exactly divisible by x2+2x-3?
You can either do a long division, and add the negative of the remainder to the original polynomial… or, rearrange the polynomial to feature as a factor x2+2x−3x2+2x−3:
x4+2x3−2x2+x−1=x4+2x3–3x2+x2+2x−3−x+2=x4+2x3−2x2+x−1=x4+2x3–3x2+x2+2x−3−x+2=
=x2(x2+2x−3)+(x2+2x−3)−x+2=(x2+1)(x2+2x−3)−x+2=x2(x2+2x−3)+(x2+2x−3)−x+2=(x2+1)(x2+2x−3)−x+2
So it looks like if you add x−2x−2 to your original polynomial, it becomes a multiple of x2+2x−
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