If 3x^3+ax^2+3x+2 leaves a reminder of 4 on division by (3x-1) find a
To find the value of "a," we will use the concept of polynomial division and the Remainder Theorem.
Step 1: Write down the polynomial division equation.
Divide 3x^3 + ax^2 + 3x + 2 by 3x - 1.
Step 2: Perform the polynomial division.
Since we want to find the remainder when dividing by 3x - 1, we need to set up the division equation as follows:
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3x - 1 | 3x^3 + ax^2 + 3x + 2
We start the division by dividing the first term of the dividend (3x^3) by the first term of the divisor (3x). This gives us x^2 as the quotient.
x^2
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3x - 1 | 3x^3 + ax^2 + 3x + 2
Next, we multiply the divisor (3x - 1) by x^2, which gives us 3x^3 - x^2. We subtract this from the original dividend.
x^2
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3x - 1 | 3x^3 + ax^2 + 3x + 2
- (3x^3 - x^2)
Simplifying, we obtain:
x^2 + ax^2 + 3x + 2
- (3x^3 - x^2)
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(a + 4)x^2 + (3 - 3a)x + 2
Step 3: Apply the Remainder Theorem.
According to the Remainder Theorem, if we divide a polynomial by 3x - 1, the remainder we obtain should be equal to the polynomial we obtained in the previous step.
Since the remainder is given as 4, we have:
(a + 4)x^2 + (3 - 3a)x + 2 = 4
Step 4: Solve for "a."
To find the value of "a" that satisfies the equation, we can rearrange the terms and equate the coefficients of each power of x.
From the equation we obtained, we can see that:
Coefficient of x^2: a + 4 = 0
Coefficient of x: 3 - 3a = 0
Solving the first equation, we have:
a + 4 = 0
a = -4
Substituting -4 into the second equation, we have:
3 - 3(-4) = 0
3 + 12 = 0
15 = 0 (which is not possible)
Therefore, there is no value of "a" that satisfies the equation while leaving a remainder of 4 when dividing 3x^3 + ax^2 + 3x + 2 by 3x - 1.
By the Remainder Theorem, you know that
f(1/3) = 4
so plug that in and find a