If (x4 + ax3 – 7x2 – 8x + b) is completely divisible by (x2 + 5x +

1.	If (x4 + ax3 – 7x2 – 8x + b) is completely divisible by (x2 + 5x + 6). Then the values of a and b are.................A) a = 2, b = 8B) a = -2, b = 6C) a = 2, b = 12D) a = 2, b = 14
Answer» Correct option is (C) a = 2, b = 12 Given that polynomial \((x^4+ax^3-7x^2-8x+b)\) is completely divisible by \((x^2+5x+6).\) If implies that \((x^2+5x+6)\) is a factor of \((x^4+ax^3-7x^2-8x+b).\) \(\Rightarrow\) (x + 2) & (x + 3) are factors of \((x^4+ax^3-7x^2-8x+b)\) \(\Rightarrow\) x = -2 & x = -3 are zeros of polynomial \((x^4+ax^3-7x^2-8x+b)\) \(\therefore(-2)^4+a(-2)^3-7(-2)^2-8(-2)+b=0\) and \((-3)^4+a(-3)^3-7(-3)^2-8(-3)+b=0\) \(\Rightarrow\) 16 - 8a - 28 + 16 + b = 0 and 81 - 27a - 63 + 24 + b = 0 \(\Rightarrow\) 8a - b = 4 and 27a - b = 42 \(\Rightarrow\) (27a - b) - (8a - b) = 42 - 4 \(\Rightarrow\) 19a = 38 \(\Rightarrow\) a = \(\frac{38}{19}\) = 2 \(\therefore\) b = 8a - 4 = 16 - 4 = 12 Correct option is C) a = 2, b = 12

If (x4 + ax3 – 7x2 – 8x + b) is completely divisible by (x2 + 5x + 6). Then the values of a and b are.................A) a = 2, b = 8B) a = -2, b = 6C) a = 2, b = 12D) a = 2, b = 14

Answer»

Correct option is (C) a = 2, b = 12

Given that polynomial \((x^4+ax^3-7x^2-8x+b)\) is completely divisible by \((x^2+5x+6).\)

If implies that \((x^2+5x+6)\) is a factor of \((x^4+ax^3-7x^2-8x+b).\)

\(\Rightarrow\) (x + 2) & (x + 3) are factors of \((x^4+ax^3-7x^2-8x+b)\)

\(\Rightarrow\) x = -2 & x = -3 are zeros of polynomial \((x^4+ax^3-7x^2-8x+b)\)

\(\therefore(-2)^4+a(-2)^3-7(-2)^2-8(-2)+b=0\)

and \((-3)^4+a(-3)^3-7(-3)^2-8(-3)+b=0\)

\(\Rightarrow\) 16 - 8a - 28 + 16 + b = 0 and 81 - 27a - 63 + 24 + b = 0

\(\Rightarrow\) 8a - b = 4 and 27a - b = 42

\(\Rightarrow\) (27a - b) - (8a - b) = 42 - 4

\(\Rightarrow\) 19a = 38

\(\Rightarrow\) a = \(\frac{38}{19}\) = 2

\(\therefore\) b = 8a - 4 = 16 - 4 = 12

Correct option is C) a = 2, b = 12