For a cubic polynomial function p, the graph of y = p(x) in the xy-plane has x-intercepts … — SAT

For a cubic polynomial function $p$ , the graph of $y = p(x)$ in the $xy$ -plane has $x$ -intercepts at $(1, 0)$ and $(4, 0)$ , and $x + 3$ is a factor of $p(x)$ . If $p(0) = -24$ , what is the value of $p(2)$ ?

A
-10
B
4
20Cevap
D
-20

Cevap

The correct answer is 20. By translating the factor

x + 3

and the intercepts

(1, 0)

and

(4, 0)

into roots, the polynomial can be modeled as

p(x) = a(x + 3)(x - 1)(x - 4)

. Solving for the leading coefficient using

p(0) = -24

yields

a = -2

. Substituting

x = 2

into the resulting equation

p(x) = -2(x + 3)(x - 1)(x - 4)

gives

p(2) = -2(5)(1)(-2) = 20

Adım Adım Çözüm

Identify the roots of the polynomial from the factors and intercepts.

The roots of

p(x)

are

x = -3

x = 1

, and

x = 4

A factor of

x + 3

corresponds to a root of

-3

. The

x

-intercepts at

(1, 0)

and

(4, 0)

correspond to roots at

1

and

4

Write the general factored form of the cubic polynomial.

p(x) = a(x + 3)(x - 1)(x - 4)

A cubic polynomial with known roots

r_1, r_2, r_3

can be expressed as

p(x) = a(x-r_1)(x-r_2)(x-r_3)

Use the value

p(0) = -24

to find the constant coefficient

a

a = -2

Substituting

x = 0

gives

p(0) = a(0 + 3)(0 - 1)(0 - 4) = 12a

. Since

12a = -24

, we solve to find

a = -2

Evaluate

p(2)

using the fully defined polynomial.

p(2) = 20

Substituting

x = 2

into

p(x) = -2(x + 3)(x - 1)(x - 4)

yields

p(2) = -2(2 + 3)(2 - 1)(2 - 4) = -2(5)(1)(-2) = 20

Anahtar Kavram

Using polynomial roots, factors, and given coordinates to define and evaluate polynomial functions.

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