On dividing a polynomial p(x) by x−1, the quotient obtained is (x3+4x2+2x+8) and the remainder obtained is 3. Which of the following is the polynomial p(x)?
(x3+4x2+2x+8) +3(x−1)(x3+4x2+2x+8) +3(x−1)
(x3+4x2+2x+8) −3(x−1)(x3+4x2+2x+8) −3(x−1)
{(x3+4x2+2x+8) (x−1)}+3{(x3+4x2+2x+8) (x−1)}+3
{(x3+4x2+2x+8) (x−1)}−3
Answers
Answer:
ANSWER
A polynomial p(x) is defined as
⇒p(x)=g(x)q(x)+r(x)
where g(x)= divisor ; q(x)= quotient and r(x)= remainder
∴ p(x) can be found by multiplying g(x) with q(x) & adding r(x) to the product.
(i).g(x)=(x−2); q(x)=x
2
−x+1; r(x)=4
∴p(x)=(x−2)[x
2
−x+1]+4
=x
3
−x
2
+x−2x
2
+2x−2+4
=x
3
−3x
2
+3x+2
(ii).g(x)=(x+3); q(x)=2x
2
+x+5; r(x)=3x+1
∴p(x)=(x+3)[2x
2
+x+5]+(3x+1)
=2x
3
+x
2
+5x+6x
2
+3x+15+3x+1
=2x
3
+7x
2
+11x+16
(iii).g(x)=(2x+1); q(x)=x
3
+3x
2
−x+1; r(x)=0
∴p(x)=(2x+1)[x
3
+3x
2
−x+1]+(0)
=2x
4
+6x
3
−2x
2
+2x+x
3
+3x
2
−x+1
=2x
4
+7x
3
+x
2
+x+1
(iv).g(x)=(x−1); q(x)=x
3
−x
2
−x−1; r(x)=2x−4
∴p(x)=(x−1)[x
3
−x
2
−x−1]+(2x−4)
=x
4
−x
3
−x
2
−x−x
3
+x
2
+x+1+2x−4
=x
4
−2x
3
+2x−3
(v).g(x)=(x
2
+2x+1); q(x)=x
4
−2x
2
+5x−7; r(x)=4x+12
∴p(x)=(x
2
+2x+1)[x
4
−2x
2
+5x−7]+(4x+12)
=x
6
−2x
4
+5x
3
−7x
2
+2x
5
+4x
3
+10x
2
−14x+x
4
−2x
2
+5x−7+4x+12
=x
6
−x
4
+x
3
+x
2
+2x
5
−5x+5
=x
6
+2x
5
−x
4
+x
3
+x
2
−5x+5
Hence, solve.
Answer:
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