Question

Consider a polynomial, f(x) = ax3 + bx² + x + 3. If x + 3 is a factor of f(x) and if f(x) is divided by x + 2, then we get remainder as 5. Then, find the values of a and b.


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Answer 1

Answer:-

Given:-

x + 3 is a factor of f(x) = ax³ + bx² + x + 3.

That is, when f(x) is divided by x + 3 ; it leaves remainder 0.

Using Factor Theorem;

⟹ x + 3 = 0

⟹ x = - 3

Substitute x = - 3 in the given polynomial.

⟹ f(- 3) = 0.

⟹ a ( - 3)³ + b ( - 3)² + ( - 3) + 3 = 0

⟹ a( - 27) + b (9) - 3 + 3 = 0

⟹ 9( - 3a + b) = 0

⟹ - 3a + b = 0

⟹ b = 3a -- equation (1)

Also given that,

When f(x) is divided by x + 2 , remainder obtained is 5.

⟹ x + 2 = 0

⟹ x = - 2

So,

⟹ f( - 2) = 5

⟹ a ( - 2)³ + b (- 2)² + ( - 2) + 3 = 5

⟹ a ( - 8) + 4b - 2 + 3 = 5

Substitute b = 3a from equation (1).

⟹ - 8a + 4(3a) + 1 = 5

⟹ 12a - 8a = 5 - 1

⟹ 4a = 4

⟹ a = 4/4

⟹ a = 1

Substitute the value of a in equation (1).

⟹ b = 3a

⟹ b = 3(1)

⟹ b = 3

∴ (a , b) = (1 , 3).

Answer 2

Answer:

Question :-

• Consider a polynomial,
• ${ax}^{3} + {bx}^{2} + x + 3$
• If x+3 is a factor of f (x) and if f (x) is divided by x+2 then we get the remainder as 5.Then find value of a and b .

♧$\mathcal\pink{Given:-}$

• Here given one polynomial and given x+3 is a factor of f (x) and f (x) is divided by x+2 then we will get remainder as x+2.

♧$\mathcal\pink{To prove:-}$

• The value of a and b.

♧$\mathcal\pink{Explanation:-}$

• Here by using a formula of remainder theorem we get that,

• $x + 3 = 0 =$
• $x = - 3.$
• Now here substitute the x=-3value in given polynomial we get that,

• $f( - 3) = {a( - 3})^{3} + b( { - 3})^{2} + ( - 3) + 3 = 0$

• $a( - 27) + b(9) - 3 + 3 = 0$

• $9( - 3a + b) = 0$
• $- 3a + b = 0 = b = 3a$

• Here we get first equation .

$\mathcal\pink {♧Next:-}$

• if f (x) is divided by x+2 we get remainder as 5.
• Therefore,

• $x + 2 = 0 = x = - 2$
• Applying the values in the given polynomial we get that,

• $a( { - 2})^{3} + b( { - 2})^{2} + ( - 2) + 3= 5$
• $a( - 8) + 4b - 2 + 3 = 5$

$\mathcal\pink {♧Now:-}$

• By substituting the value of b=3a

• $- 8a + 4(3a) + 1 = 5$
• $12a - 8a = 5 - 1 = 4a = 4$
• $a = \frac{4}{4} = a = 1$

$\mathcal\pink {♧Therefore:-}$

• Value of a=1
• Value of b=3×1=3

Value of a and b is (1,3).

♧Hope it helps u mate .

♧Thank you .