Question

Find the remainder when the polynomial P(x)=x^4+2x^3-3x^2+x-1 is divided by (x-2)

Answer 1

Question

Find the remainder when the polynomial P(x)=x⁴+2x³-3x²+x-1 is divided by (x-2)

Given

p(x) = x⁴+2x³-3x²+x-1

g(x) = x-2

To Find

Remainder =?

Explanation

p(x) = x⁴+2x³-3x²+x-1

g(x) = x-2

= x-2=0

= x=2

Putting x=2 in p(x)

p(x) = x⁴+2x³-3x²+x-1

= (2)⁴+2×(2)³-3×(2)²+2-1

= 16 + 2× 8-3×4+2-1

= 16+16-12+2-1

= 32-12+2-1

= 34-12-1

= 34-13

= 21

Hope it helps!

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Answer 2

The given polynomial is p(x)=x

4 −2x

3 +3x 2−ax+3a−7

Given that, the polynomial p(x) when divided by (x+1) leaves remainder 19

Therefore, p(−1)=19 (By Remainder theorem)

=>(−1)

4 −2×(−1)

3 +3(−1)

2 −(−1)a+3a−7=19

=>1+2+3+a+3a−7=19

=>4a−1=19

=>4a=20

=>a=5

The value of a is 5

Now,

p(x)=x

4 −2x

3+3x

2−5x+3×5−7

=x 4 −2x

3 +3x

2 −5x+15−7

=x 4 −2x

3+3x

2 −5x+8

Remainder when the polynomial is divided by (x+2)

=p(−2) (By Remainder Theorem)

=−2

4 −2(−2)

3+3(−2)

2 −5(−2)+8

=16+16+12+10+8

=62

Thus, the remainder of the polynomial p(x) when divided by (x+2) is 62