A polynomial f(x) is defined by f(x) = 2x^3 + hx^2 - 7x - 6, where h is a constant. If the… — SAT

A polynomial $f(x)$ is defined by $f(x) = 2x^3 + hx^2 - 7x - 6$ , where $h$ is a constant. If the graph of $y = f(x)$ in the $xy$ -plane passes through the point $(2, 0)$ , what is the value of $h$ ?

Answer: 1

Answer

The value of the constant

h

is 1.

Since the graph of

y = f(x)

passes through the point

(2, 0)

, the value of the function at

x = 2

must be

0

. Substituting

x = 2

into the equation yields

2(2)^3 + h(2)^2 - 7(2) - 6 = 0

. Simplifying the terms gives

2(8) + 4h - 14 - 6 = 0

, which simplifies to

16 + 4h - 20 = 0

, or

4h - 4 = 0

. Solving for

h

yields

h = 1

Step-by-Step Solution

Apply the point condition to the polynomial function.

f(2) = 0

Since the point

(2, 0)

lies on the graph of

y = f(x)

, substituting

x = 2

must yield

y = 0

Substitute

x = 2

into the expression for

f(x)

f(2) = 2(2)^3 + h(2)^2 - 7(2) - 6

We replace each occurrence of

x

with

2

to evaluate the function value.

Simplify the algebraic expression.

16 + 4h - 20 = 0 \implies 4h - 4 = 0

Evaluating the exponents and multiplications gives

2(8) + 4h - 14 - 6 = 16 + 4h - 20 = 4h - 4

Solve the linear equation for

h

h = 1

Adding 4 to both sides gives

4h = 4

, and dividing by 4 gives

h = 1

Key Concept

Connecting graphical x-intercepts of a polynomial to its algebraic roots and evaluation

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