Question

2 and 4 sum in photo

2- p(x)=x⁴-5x²+7x³-3x+2; q(x)= x-1
using factor theorem decide weather q(x) is factor of p(x) or not

step by step explanation:

4 sum in photo
y³-3y²+3y-1 ÷y+1

step by step explaination:

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Answer 1

Question 1) :- p(x)=x⁴-5x²+7x³-3x+2; q(x)= x-1

using factor theorem decide weather q(x) is factor of p(x) or not ?

Solution :-

we know that, factor theorem says that, a polynomial f(x) has a factor (x - a) , if and only if f(a) = 0.

So, putting q(x) = 0, we get,

→ x - 1 = 0

→ x = 1.

checking in p(x) now,

→ p(x)=x⁴-5x²+7x³-3x+2

→ p(1) = (1)⁴ - 5*(1)² + 7(1)³ - 3*1 + 2

→ p(1) = 1 - 5*1 + 7*1 - 3 + 2

→ p(1) = 1 - 5 + 7 - 3 + 2

→ p(1) = 1 + 7 + 2 - 5 - 3

→ p(1) = 10 - 8

→ p(1) = 2.

we get,

→ p(1) ≠ 0.

Therefore, we can say that, q(x) is not a factor of p(x).

Question 2) :- Find the remainder using remainder theorem :- (y³ - 3y² + 3y - 1) ÷ (y + 1).

Solution :-

we know that, remainder theorem says that, when any polynomial f(x) is divided by (x + a) , than the remainder will be f(-a).

So,

→ y + 1 = 0

→ y = (-1)

Putting y = (-1) , we get,

→ f(y) = (y³ - 3y² + 3y - 1)

→ f(-1) = (-1)³ - 3(-1)² + 3(-1) - 1

→ f(-1) = (-1) - 3*1 - 3 - 1

→ f(-1) = (-1) - 3 - 4

→ f(-1) = (-8) (Ans.)

Hence, the remainder will be (-8).