Suppose that P(x) is a linear polynomial with P(6)-P(2)=12. What is P(12)-P(2)?
[(12 - 2) / (6 - 2) ] * 12
30
To find the value of P(12) - P(2), we can use the given information and properties of linear polynomials.
Let's start by using the given equation: P(6) - P(2) = 12.
Since P(x) is a linear polynomial, we can rewrite it as P(x) = mx + b, where m represents the slope and b represents the y-intercept.
Let's substitute the given values into the equation using x = 6 and x = 2:
P(6) = m(6) + b
P(2) = m(2) + b
Rewriting the equations:
m(6) + b - (m(2) + b) = 12
Simplifying the equation:
6m + b - 2m - b = 12
4m = 12
Divide both sides by 4:
m = 3
Now that we have determined the slope, we can substitute the value of m back into one of the earlier equations to find the y-intercept, b.
Using P(2) = m(2) + b:
P(2) = 3(2) + b
P(2) = 6 + b
Since we already know that P(2) - P(2) = 0, we can rewrite the equation as:
0 = 6 + b
Simplifying the equation:
b = -6
Now that we have determined the values of m and b, we can write the linear polynomial equation:
P(x) = 3x - 6
To find P(12) - P(2), we substitute 12 and 2 into the equation:
P(12) = 3(12) - 6 = 36 - 6 = 30
P(2) = 3(2) - 6 = 6 - 6 = 0
Therefore, P(12) - P(2) is equal to 30 - 0 = 30.