The function f is defined by f(x) = x^3 - 3x^2 - 10x + k, where k is a constant. In the xy… — SAT

The function $f$ is defined by $f(x) = x^3 - 3x^2 - 10x + k$ , where $k$ is a constant. In the $xy$ -plane, the graph of $y = f(x)$ has $x$ -intercepts at $(c, 0)$ and $(2c, 0)$ , where $c$ is a positive constant. What is the value of $k$ ?

Answer: 24

Answer

By applying the factor theorem to the two

x

-intercepts

(c,0)

and

(2c,0)

, we establish the system

c^3 - 3c^2 - 10c + k = 0

and

8c^3 - 12c^2 - 20c + k = 0

. Subtracting these equations yields

7c^3 - 9c^2 - 10c = 0

. Since

c > 0

, dividing by

c

gives

7c^2 - 9c - 10 = 0

, which factors as

(7c+5)(c-2)=0

, yielding the positive solution

c=2

. Substituting

c=2

back into the first equation gives

8 - 12 - 20 + k = 0

, which solves to

k = 24

Step-by-Step Solution

Set up equations for the roots

c

and

2c

using the factor theorem.

f(c) = c^3 - 3c^2 - 10c + k = 0

and

f(2c) = 8c^3 - 12c^2 - 20c + k = 0

x

-intercept at

(r, 0)

means that

r

is a root of the polynomial, so

f(r) = 0

Subtract the first equation from the second to eliminate the constant

k

7c^3 - 9c^2 - 10c = 0

Eliminating

k

allows us to solve for the root

c

directly.

Solve the polynomial equation for the positive constant

c

c = 2

Dividing the equation by

c

(since

c > 0

) yields

7c^2 - 9c - 10 = 0

, which factors as

(7c+5)(c-2) = 0

. Since

c

must be positive,

c = 2

Substitute

c = 2

back into the equation for

f(c) = 0

to solve for

k

k = 24

Substituting the known root value allows us to find the value of the constant coefficient

k

Key Concept

The relationship between a polynomial's algebraic factors, its roots, and its

x

-intercepts in the coordinate plane.

Estimated Time:2m 0s

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