Question

7. Find 'a' if the two polynomials ax³ + 3x² - 9
and 2x³ + 4x + a, leave the same remainder
when divided by x + 3​

Answer 1

Answer: a = 3

Given three polynomials:

• p(x) = ax³ + 3x² - 9
• f(x) = 2x³ + 4x + a
• g(x) = x + 3

Also, Polynomials p(x) & f(x) leaves the same reminder when divided by g(x), This can be expressed as:

⇒ p( g(x) ) = f( g(x) ) ...(i)

[ Remainder Theorem ]

So, Let's find the factor of g(x), Because a factor turns the given polynomial into a zero polynomial.

Hence, Equating g(x) with 0 , we will get the only factor ( Since, g(x) is a linear polynomial)

⇒ g(x) = 0

⇒ x + 3 = 0

⇒ x = -3

As discussed earlier, Both polynomials leaves the same reminder when divided by g(x) { See eq.(1) }

⇒ p(-3) = g(-3)

⇒ a(-3)³ + 3(-3)² - 9 = 2(-3)³ + 4(-3) + a

⇒ -27a + 27 - 9 = -54 - 12 + a

⇒ 27 - 9 + 54 + 12 = a + 27a

⇒ 28a = 18 + 12 + 54

⇒ 28a = 30 + 54

⇒ 28a = 84

⇒ a = 3

Hence, Value of a is 3.

Zero Polynomial:

A polynomial that has all its coefficients 0.

e.g., 0x² + 0x + C

[ C = constant ]

Answer 2

Answer:

$\bf a=\frac{84}{28} =3$

Step-by-step explanation:

Let p(x)=ax³+3x²-9

and g(x)=2x³+4x+a

As per remainder theorem the remainder when p(x)

is divided by x+3 ( x=-3)

R= p(-3)

$R=a(-3)^3+3(-3)^2-9\\\\R= - 27a+27-9= -27a+18--------------(1)$

and the remainder when g(x)

is divided by x+3 ( x=-3)

$R'=g(-3)\\\\=2(-3)^3+4(-3)+a\\\\\bf{ R'=-54-12+a=-66+a-----------(2)}$

$Given \ that \ R=R'\\\\-27a+18=-66+a\\\\28a=18+66=84\\\\\bf a=\frac{84}{28} =3$