the expression 6x^2-2x+3 leaves a reminder of 3 when divided by x-p,Determine the value of p
6p^2-2p+3 = 3
6p^2 - 2p = 0
2p(3p-1) = 0
p=0 or p = 1/3
please, I want to know math
To determine the value of p, we need to use the Remainder Theorem.
The Remainder Theorem states that if a polynomial f(x) is divided by (x - p), the remainder is equal to f(p).
In this case, the expression 6x^2 - 2x + 3 leaves a remainder of 3 when divided by (x - p). Therefore, we can set up the equation:
6p^2 - 2p + 3 = 3
Simplifying the equation, we have:
6p^2 - 2p = 0
Now, we can factor out a common factor of 2p:
2p(3p - 1) = 0
Setting each factor equal to zero, we have:
2p = 0 or 3p - 1 = 0
From the first equation, p = 0. From the second equation, we solve for p:
3p - 1 = 0
3p = 1
p = 1/3
Therefore, the value of p is either 0 or 1/3.
To determine the value of p, we need to set up an equation using the given information.
Given that the expression 6x^2 - 2x + 3 leaves a remainder of 3 when divided by x - p, we can express this as follows:
(6x^2 - 2x + 3) / (x - p) = 3
Now, let's divide the quadratic expression by the binomial using long division:
6x + 12
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x - p | 6x^2 - 2x + 3
When we perform long division, the quotient is 6x + 12. The remainder should be zero, but it is given as 3. Therefore, we can set up the equation:
(6x^2 - 2x + 3) - (6x + 12)(x - p) = 3
Expanding the expression:
6x^2 - 2x + 3 - (6x^2 - 6px + 12x - 12p) = 3
Combine like terms:
6x^2 - 2x + 3 - 6x^2 + 6px - 12x + 12p = 3
Group like terms:
(6px - 12x) + (6x^2 - 6x^2) + (12p + 3 - 2x) = 3
Simplify:
6px - 12x + 12p + 3 - 2x = 3
Rearrange the terms:
6px - 12x - 2x + 12p + 3 - 3 = 0
Combine like terms:
6px - 14x + 12p = 0
Now, if this equation holds for all values of x, the coefficients of x and the constant terms must be zero:
-14x = 0 (coefficient of x)
6px + 12p = 0 (constant terms)
From the equation -14x = 0, we can determine x = 0.
Plugging x = 0 into the second equation:
6p(0) + 12p = 0
0 + 12p = 0
12p = 0
p = 0
Therefore, the value of p is 0.