(4)When polynomial x^3 + 3x^2 + 3x + 1 is divided by x + 1, the remain

1.	(4)When polynomial x^3 + 3x^2 + 3x + 1 is divided by x + 1, the remainder is_____(a) 1(b) 0(c) 3(d) -6(5)If Y coordinate of a point is zero, then this point always lies in_______(a) Quadrant 1(b) Quadrant 2 (c) On x-axis (d) On y-axis
Answer» Answer: p.folow me Step-by-step explanation: When a polynomial p(x) is divided by (x−a) , the remainder is p(a) .” Also, when a polynomial p(x) is divided by another polynomial q(x) ,the degree of the remainder is at most 1 less than the degree of q(x) . USING the remainder THEOREM, we can write: p(1)=3;p(3)=5 p(x) can be written as: Dividend=(Divisor×Quotient)+Remainder p(x)=(x−1)(x−3)Q(x)+r(x) r(x) is the remainder polynomial and Q(x) is the Quotient polynomial. Since r(x) is linear, r(x)=Ax+B , where A and B are ARBITRARY constants. ⟹p(x)=(x−1)(x−3)Q(x)+Ax+B Now, p(1)=A+B=3⟹A+B=3 p(3)=3A+B=5⟹3A+B=5 Solving the 2 EQUATIONS, we get A=1 and B=2 . Therefore, r(x)=Ax+B=x+2 r(−2)=−2+2=0 Hence, r(−2)=0

(4)When polynomial x^3 + 3x^2 + 3x + 1 is divided by x + 1, the remainder is_(a) 1(b) 0(c) 3(d) -6(5)If Y coordinate of a point is zero, then this point always lies in___(a) Quadrant 1(b) Quadrant 2 (c) On x-axis (d) On y-axis

Answer»

Answer:

p.folow me

Step-by-step explanation:

When a polynomial p(x) is divided by (x−a) , the remainder is p(a) .”

Also, when a polynomial p(x) is divided by another polynomial q(x) ,the degree of the remainder is at most 1 less than the degree of q(x) .

USING the remainder THEOREM, we can write:

p(1)=3;p(3)=5

p(x) can be written as:

Dividend=(Divisor×Quotient)+Remainder

p(x)=(x−1)(x−3)Q(x)+r(x)

r(x) is the remainder polynomial and Q(x) is the Quotient polynomial. Since r(x) is linear, r(x)=Ax+B , where A and B are ARBITRARY constants.

⟹p(x)=(x−1)(x−3)Q(x)+Ax+B

Now,

p(1)=A+B=3⟹A+B=3

p(3)=3A+B=5⟹3A+B=5

Solving the 2 EQUATIONS, we get A=1 and B=2 .

Therefore, r(x)=Ax+B=x+2

r(−2)=−2+2=0

Hence, r(−2)=0