Question 180·Hard·Nonlinear Functions
The function
is defined above, where , , and are real constants. The function has a zero at of multiplicity 2. If , what is the value of ?
(Express the answer as an integer)
When a polynomial question involves a root with multiplicity greater than 1, immediately rewrite the polynomial in factored form using that repeated factor, introducing a simple variable (like r) for any unknown root. Then use the given function value (such as f(1) = 6) to form and solve a basic equation for that root. Finally, use shortcut evaluations like f(0) = c to get the constant term directly, avoiding full expansion unless absolutely necessary; this keeps the algebra and arithmetic quick and clean on the SAT.
Hints
Translate multiplicity into a factor
A zero at x = 2 with multiplicity 2 means (x - 2) appears how many times as a factor of f(x)? Write f(x) in terms of (x - 2)² and one more linear factor.
Introduce an unknown for the remaining root
Call the remaining root r and write f(x) in the form (x - 2)²(x - r). Then plug x = 1 into this expression and use the given value of f(1).
Use the definition of c
Remember that for f(x) = x³ + ax² + bx + c, the constant term c equals f(0). Once you know the factored form, plug in x = 0 and simplify to get c.
Desmos Guide
Represent the factored form with an unknown root
In Desmos, define a parameter r and enter the function
f(x) = (x - 2)^2 (x - r)
Add a slider for r if Desmos offers it automatically.
Use f(1) = 6 to determine r
In a new expression line, enter (1 - 2)^2 * (1 - r) and then enter y = 6 on another line. Plot y = (1 - 2)^2 * (1 - r) as a function of r (Desmos will treat r as the variable) and find the intersection with the horizontal line y = 6. The r-coordinate of this intersection is the remaining root.
Compute c from the remaining root
Once you know r, recall that c = f(0). In Desmos, type (0 - 2)^2 * (0 - r) or equivalently c = (0 - 2)^2 * (0 - r). The numeric value Desmos shows for this expression is the value of c you should report.
Step-by-step Explanation
Use the multiplicity information to factor f(x)
If a polynomial has a zero (root) at x = 2 of multiplicity 2, that means (x - 2)² is a factor.
So we can write the cubic in factored form as
for some real number r (the third root).
Use f(1) = 6 to find the third root
Substitute x = 1 into the factored form of f(x):
Compute (1 - 2)²:
So
We are told f(1) = 6, so set up the equation
and solve for r.
Solve for r
From
subtract 1 from both sides:
Multiply both sides by -1:
So the fully factored form is
Find c using f(0) = c
By definition,
so when x = 0,
Use the factored form to compute f(0):
Therefore, the value of is 20.