3x³-5x²+8x-9÷x-1 slove this
Answers
Given 3x³ + 5x² - 3x - 5 = 0………………………………………(1)
Equation (1) is a cubic in one variable x . Therefore, it will have three solutions for x which we will get by method of factorization. Rewriting (1)→
(3x³ - 3x) + (5x² - 5) = 0
Taking the common factors 3x and 5 outside the brackets,
3x(x² - 1) + 5(x² - 1) = 0
Or, (x² - 1) (3x + 5) = 0
Now the left-hand-side is a product of two factors (x² - 1) and (3x + 5), and the right-hand-side is zero . Therefore if either of the two factors is zero, the right-side will be zero. This means,
either (x² - 1) = 0 or (3x + 5) = 0
If x² - 1 = 0, by transposing -1 to right,
x² = 1
⇒ x = -1, 1 ………………………………………………..…………..(2)
From 3x + 5 = 0, by transposition
3x = -5
On dividing both sides by 3,
3x/3 = -5/3
Or, 1.x = -5/3
⇒ x = -5/3 ………………………………………………………. ….(3)
From (2)and (3), we get the required three values of x :
x = -1, 1, -5/3
Verification:
Put x=1 in (1) to get 3.1 + 5.1 - 3.1 - 5 = 3+5–3–5 = 8–8 = 0
Put x=-1 in (1) to get 3(-1)³ + 5(-1)² - 3(-1) - 5 = -3+5+3–5=0
Put x = -5/3 in (1) to get
3(-5/3)³ + 5(-5/3)² - 3(-5/3) - 5 = -3.125/27 + 125/9 +5 - 5= -125/9 + 125/9 +0
= 0 + 0 + 0 = 0
∴ all the three values of x satisfy the equation (3).
Hence x = -1, 1, -5/3 are the correct solutions.
Answer:
Quotient= 3x²-2x+6 and remainder= (-3).
Step-by-step explanation:
Given⤵
➡3x³-5x+8x-9 divided by x-1.
To find ⤵
➡Value after dividing.
Solution⤵
x-1) 3x³-5x²+8x-9(3x²-2x+6
3x³-3x²
- +
___________
-2x²+8x
-2x²+2x
+ -
_________
6x-9
6x-6
- +
_________
-3
✅Hence,uotient = 3x²-2x+6 and remainder is -3.
Proof ⤵
Dividend= Quotient×Divisor+Remainder
➡(3x²-2x+6)×(x-1)+(-3)
➡x(3x²-2x+6)-1(3x²-2x+6)+(-3)
➡3x³-2x²+6x-3x²+2x-6-3
➡3x³-2x²-3x²+6x+2x-9
➡3x³-5x²+8x-9= LHS
Hope this is helpful to you!