The function f is defined by f(x) = (x - 3)(x^3 - kx^2 + 5x - 15), where k is a constant. … — SAT

The function $f$ is defined by $f(x) = (x - 3)(x^3 - kx^2 + 5x - 15)$ , where $k$ is a constant. In the $xy$ -plane, the graph of $y = f(x)$ is tangent to the $x$ -axis at the point $(3, 0)$ . What is the value of $k$ ?

Cevap: 3

Cevap

For the graph of a polynomial function to be tangent to the

x

-axis at

(3, 0)

, the root

x = 3

must have an even multiplicity (at least 2). The function is defined as

f(x) = (x - 3)(x^3 - kx^2 + 5x - 15)

. Since there is already one factor of

(x - 3)

explicitly defined, the remaining cubic factor

g(x) = x^3 - kx^2 + 5x - 15

must also have a factor of

(x - 3)

to make the total multiplicity of the root

x = 3

at least 2. According to the Factor Theorem, if

(x - 3)

is a factor of

g(x)

, then

g(3) = 0

. Substituting

x = 3

into

g(x)

gives

3^3 - k(3)^2 + 5(3) - 15 = 0

, which simplifies to

27 - 9k = 0

. Solving this equation for

k

yields

k = 3

Adım Adım Çözüm

Identify the relationship between graph tangency and factor multiplicity.

For the graph of a polynomial function to be tangent to the

x

-axis at a point

(c, 0)

, the factor

(x - c)

must have an even multiplicity of at least 2 in the polynomial's factored form.

An odd multiplicity root causes the graph to cross the

x

-axis, while an even multiplicity root causes the graph to touch the

x

-axis and turn around (tangency).

Apply the multiplicity requirement to the given function.

Since

f(x) = (x - 3)(x^3 - kx^2 + 5x - 15)

already contains one factor of

(x - 3)

, the cubic expression

g(x) = x^3 - kx^2 + 5x - 15

must also contain

(x - 3)

as a factor to ensure the total multiplicity of the root

x = 3

is at least 2.

This guarantees that

(x - 3)^2

is a factor of

f(x)

Apply the Factor Theorem to the cubic expression.

Since

(x - 3)

is a factor of

g(x)

, then

g(3) = 0

The Factor Theorem states that a polynomial

P(x)

has a factor

(x - c)

if and only if

P(c) = 0

Solve for the constant

k

by substituting

x = 3

into

g(x)

3^3 - k(3)^2 + 5(3) - 15 = 0 \implies 27 - 9k + 15 - 15 = 0 \implies 27 - 9k = 0 \implies 9k = 27 \implies k = 3

Arithmetic simplification yields the value of the constant.

Anahtar Kavram

The relationship between polynomial factors, root multiplicities, and the behavior of the graph at

x

-intercepts.

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