if x3+ax2-bx+10 is divisible by x2-3x+2 .find the value of a and b
To find the values of a and b, we can use polynomial division to divide x^3 + ax^2 - bx + 10 by x^2 - 3x + 2.
Let's perform the division:
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x^2 - 3x + 2 | x^3 + ax^2 - bx + 10
To begin the division, we divide x^3 by x^2, which gives us x. Now, we multiply (x^2 - 3x + 2) by x:
x(x^2 - 3x + 2)
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x^2 - 3x + 2 | x^3 + ax^2 - bx + 10
-(x^3 - 3x^2 + 2x)
Subtracting the result from the previous step from the dividend, we get:
4x^2 - bx + 10
_________________________
x^2 - 3x + 2 | x^3 + ax^2 - bx + 10
-(x^3 - 3x^2 + 2x)
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-ax^2 - bx + 10
Next, we divide -ax^2 by x^2, which gives us -a. Now, we multiply (x^2 - 3x + 2) by -a:
-a(x^2 - 3x + 2)
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x^2 - 3x + 2 | x^3 + ax^2 - bx + 10
-(x^3 - 3x^2 + 2x)
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-ax^2 - bx + 10
ax^2 - 3ax + 2a
Subtracting the result from the previous step from the current remainder, we get:
(-4a - b)x + (10 + 2a)
Since the polynomial division states that the remainder is 0, we can equate the above expression to zero:
(-4a - b)x + (10 + 2a) = 0
This equation holds for all x. In order for this equation to hold for any x, the coefficient of x and the constant term must be zero:
-4a - b = 0 (Equation 1)
10 + 2a = 0 (Equation 2)
From Equation 2, we can solve for a:
2a = -10
a = -5
Substituting the value of a in Equation 1, we can solve for b:
-4(-5) - b = 0
20 - b = 0
b = 20
Therefore, the values of a and b are a = -5 and b = 20, respectively.
To check if one polynomial is divisible by another polynomial, you can use polynomial long division or synthetic division. Let's use polynomial long division in this case.
Step 1: Write the dividend (the polynomial being divided) and the divisor:
Dividend: x3 + ax2 - bx + 10
Divisor: x2 - 3x + 2
Step 2: Divide the highest-degree term of the dividend by the highest-degree term of the divisor: x3 divided by x2 gives x.
x
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x2 - 3x + 2 | x3 + ax2 - bx + 10
Step 3: Multiply the divisor by the quotient obtained in Step 2 and subtract the result from the dividend:
(x2 - 3x + 2) * x = x3 - 3x2 + 2x
x3 + ax2 - bx + 10
- (x3 - 3x2 + 2x)
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ax2 - (b + 3)x + (10 - 2x)
Step 4: Repeat Steps 2 and 3 with the new dividend:
Dividend: ax2 - (b + 3)x + (10 - 2x)
Now, we need to check if the new dividend ax2 - (b + 3)x + (10 - 2x) is divisible by the divisor x2 - 3x + 2.
For the new dividend to be divisible by the divisor, the remainder has to be zero. In other words, we need the expression:
ax2 - (b + 3)x + (10 - 2x)
to equal zero.
To zero out this expression, we match the coefficients of each power of x with zero. This gives us the following equations:
a = 0 (since there is no x2 term in the expression)
b + 3 = 0 (from the x coefficient)
10 - 2x = 0 (from the constant term)
From the last equation, we can find the value of x:
10 - 2x = 0
2x = 10
x = 5
Now, we substitute the value of x into the second equation to solve for b:
b + 3 = 0
b = -3
Therefore, the value of a is 0 and the value of b is -3.
good
since x^2 - 3x + 2 = (x-2)(x-1)
so x^3 + ax^2 -bx + 10 must be divisible by both
x-2 and x-1
let f(x) = x^3 + ax^2- bx + 10
f(2) = 5 + 4a - 2b + 10 = 0 ---> 4a - 2b = -15
f(1) = 1 + a - b + 10 ---> a-b = -11 or a = b-11
sub 2nd into 1st
4(b-11) - 2b = -15
4b - 44 - 2b = -15
2b = 29
b = 29/2
then a = 29/2 - 11 = 7/2