Question

Find the value of a and b so that x3 - ax2 - 13x + b has x+1 and x+3 as factors

Answer 1

Polynomial , x³ -ax² -13x + b has two factors ( x -1) and (x +3)

it means , x = 1 and -3 are the roots (zeros ) of x³ -ax² -13x + b .

so,

put x = 1 in polynomial ,.

(1)³ -a(1)² -13(1) + b = 0

1 - a -13 + b = 0

-a + b = 12 ---------(1)

put x = -3

(-3)³ -a(-3)² -13(-3) + b = 0

-27 -9a +39 + b = 0

-9a + b = -12 --------(2)

subtract equation (1) and (2)

-a + b + 9a - b = 12 + 12

8a = 24

a = 3

put a = 3 in equation (1)

b = 15

Answer 2

$\huge{\frak{\underline{Solution:-}}}$

As, (x - 1) and (x + 3) are the factors of x³- ax² - 13x + b, then:

• x - 1 = 0
• x = 1

Putting x = 1 in the equation:-

$\rightarrow$ x³- ax² - 13x + b = 0

$\rightarrow$ (1)³- a(1)² - 13(1) + b = 0

$\rightarrow$ 1 - a - 13 + b = 0

$\rightarrow$ -12 - a + b = 0

$\rightarrow$ - a + b = 12 - - - - - - (i)

Now,

• x + 3 = 0
• x = -3

Putting x = -3 in the equation:-

$\rightarrow$ x³- ax² - 13x + b = 0

$\rightarrow$ (-3)³- a(-3)² - 13(-3) + b = 0

$\rightarrow$ -27 - 9a + 39 + b = 0

$\rightarrow$ 12 - 9a + b = 0

$\rightarrow$ - 9a + b = -12 - - - - - - (ii)

On subtracting equation (i) and (ii)

= - a + b - ( - 9a + b ) = 12 - (- 12)

= - a + b + 9a - b = 12 + 12

= 8a = 24

= a = 24/8

= a = 3

Put a = 3 in the equation to find the value of b:

= -a + b = 12

= -(3) + b = 12

= b = 12 + 3

= b = 15

• Thus, the value of a=3 and b=15.

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