Question

. When a polynomial 2x +3x² + ax + b is divided by (x - 2) leaves remainder 2, and
(x+2) leaves remainder -2. Find a and b.​

Answer 1

SOLUTION:-

$\bf\underline{\underline{\red{Question:-}}}$

⠀⠀⠀⠀⠀• When a polynomial 2x +3x² + ax + b is ⠀⠀⠀⠀⠀divided by (x - 2) leaves remainder 2, and

⠀⠀⠀⠀⠀(x+2) leaves remainder -2. Find a and b?

$\bf\underline{\underline{\orange{Answer:-}}}$

⠀⠀⠀⠀⠀• a= -11

⠀⠀⠀⠀⠀• b= -3/2

$\bf\underline{\underline{\green{Given:-}}}$

⠀⠀⠀⠀⠀• P(x) = 2x+3x²+ax+b

⠀⠀⠀⠀⠀• g(x) = (x+2),(x-2)

$\bf\underline{\underline{\purple{To\: Calculate:-}}}$

⠀⠀⠀⠀⠀• Value of a and b

$\bf\underline{\underline{\blue{Explanation:-}}}$

• p(x) = 2x+3x²+ax+b

⇝ 3x²+2x+ax+b

g(x) = x-2

⇝ x-2=0

⇝ x=2

Case 1

When a polynomial 2x +3x² + ax + b is divided by (x - 2) leaves remainder 2

Putting value of x in p(x)

p(x) = 3x²+2x+ax+b =2

➫ 3×(2)²+2×(2)+a×2+b=2

➫ 3×4+4+2a+b=2

➫ 12+4+2a+b=2

➫ 16+2a+b=2

➫2a+b=-14_______________(1)

Case2

When a polynomial 2x +3x² + ax + b is divided by (x +2) leaves remainder -2

p(x) = 3x²+2x+ax+b=0

g(x) = x+2

⇝ x=-2

Putting value of x in p(x)

➫ 3×(-2)²+2×(-2)+a×(-2)+b=0

➫ 3×4-4-2a+b=0

➫ 12-4-2a+b=0

➫ -2a+b=-8____________(2)

On Adding Eq1 and Eq2

-2a + b = -8

2a + b = -14

__________

2b = -22

b= -22/2

${\boxed{\red{\tt{b= -11 }}}}$

putting value of b in eq 1

2a+b=-14

⇢ 2a+(-11)=-14

⇢2a-11=-14

⇢ 2a=-14+11

⇢ 2a= -3

${\boxed{\red{\tt{a= -3/2 }}}}$

$\bf\underline{\underline{\red{Hence:-}}}$

⠀⠀⠀⠀⠀• a= -11

⠀⠀⠀⠀⠀• b= -3/2

_____________________________________________

Answer 2

Answer:

Given:−

⠀⠀⠀⠀⠀• P(x) = 2x+3x²+ax+b

⠀⠀⠀⠀⠀• g(x) = (x+2),(x-2)

\bf\underline{\underline{\purple{To\: Calculate:-}}}

ToCalculate:−

⠀⠀⠀⠀⠀• Value of a and b

\bf\underline{\underline{\blue{Explanation:-}}}

Explanation:−

• p(x) = 2x+3x²+ax+b

⇝ 3x²+2x+ax+b

g(x) = x-2

⇝ x-2=0

⇝ x=2

Case 1

When a polynomial 2x +3x² + ax + b is divided by (x - 2) leaves remainder 2

Putting value of x in p(x)

p(x) = 3x²+2x+ax+b =2

➫ 3×(2)²+2×(2)+a×2+b=2

➫ 3×4+4+2a+b=2

➫ 12+4+2a+b=2

➫ 16+2a+b=2

➫2a+b=-14_______________(1)

Case2

When a polynomial 2x +3x² + ax + b is divided by (x +2) leaves remainder -2

p(x) = 3x²+2x+ax+b=0

g(x) = x+2

⇝ x=-2

Putting value of x in p(x)

➫ 3×(-2)²+2×(-2)+a×(-2)+b=0

➫ 3×4-4-2a+b=0

➫ 12-4-2a+b=0

➫ -2a+b=-8____________(2)

On Adding Eq1 and Eq2

-2a + b = -8

2a + b = -14

__________

2b = -22

b= -22/2

{\boxed{\red{\tt{b= -11 }}}}

b=−11

putting value of b in eq 1

2a+b=-14

⇢ 2a+(-11)=-14

⇢2a-11=-14

⇢ 2a=-14+11

⇢ 2a= -3

{\boxed{\red{\tt{a= -3/2 }}}}

a=−3/2

\bf\underline{\underline{\red{Hence:-}}}

Hence:−

⠀⠀⠀⠀⠀• a= -11

⠀⠀⠀⠀⠀• b=3/2

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