Question

the polynomial p(x)=x4-2x3+3x2-ax+3a-7 when divided by (x+1) leaves remainder 19.Find the values of a.Also find the remainder when p(x) is divided by (x+3) pls help

Answer 1

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Answer 2

Question:

The polynomial p(x) = x⁴-2x³+3x²-ax+3a-7 when divided by ( x+1 ) leaves remainder 19. Find the value of a . Also find the remainder when p(x) is divided by ( x+3 ).

Answer:

➛Value of a : 5

➛Remainder when p(x) is divided by ( x+3 ) : 68

Given:

The given polynomial is :

➞ p(x) = x⁴-2x³+3x²-ax+3a-7

To Find :

1) The value of a

2) Remainder when p(x) is divided by ( x+3 )

Solution:

The given polynomial is :

➞ p(x) = x⁴-2x³+3x²-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 - ax + 3( -1 ) - 7 = 19

➡ 1 + 2 + 3 + a + 3a - 7 = 19

➡ 4a - 1 = 19

➡ 4a = 20

➡ a = 5

Now,

p(x) = x⁴ - 2x³ + 3x² - 5x + 3( 5 ) - 7

p(x) = x⁴ - 2x³ + 3x² - 5x + 8

Now, To find

Remainder when the polynomial is divided by ( x+3 ):

p( -3 ) (By Remainder Theorem)

p( -3 ) = ( -3 )⁴ - 2( -3 )³ + 3( -3 )² - 5( -3 ) + 8

p( -3 ) = 81 - 2( -27 ) + 3( 6 ) + 15 + 8

p( -3 ) = 81 - 54 + 18 + 15 + 8

p( -3 ) = 68

Thus, the remainder of the polynomial p(x) when divided by (x+3) is 68