Find the values of a and b, if (x - 1) and (x + 3) are the factors of x³- ax² - 13x + b
Answers
Step-by-step explanation:
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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- 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:-
→ x³- ax² - 13x + b = 0
→ (1)³- a(1)² - 13(1) + b = 0
→ 1 - a - 13 + b = 0
→ -12 - a + b = 0
→ - a + b = 12 - - - - - - (i)
Now,
x + 3 = 0
x = -3
Putting x = -3 in the equation:-
→ x³- ax² - 13x + b = 0
→ (-3)³- a(-3)² - 13(-3) + b = 0
→ -27 - 9a + 39 + b = 0
→ 12 - 9a + b = 0
→ - 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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