Question

The polynomial p is graphed. graph of cubic polynomial p that passes through the points: negative 3, 2; negative 2, negative 4; and 2, 0.what is the remainder when p(x) is divided by (x 2)?

Answer 1

The Remainder Theorem is used to find the remainder of a polynomial divided by (x - 2). Since p(2) = 0 for the polynomial p(x), the remainder is zero.

Step-by-step explanation:

The student asks a question about a polynomial function p(x) and seeks to understand the remainder when this polynomial is divided by (x - 2). To find the remainder when dividing a polynomial by a linear factor, such as (x - 2), one can use the Remainder Theorem. According to the Remainder Theorem, if a polynomial f(x) is divided by (x - k), then the remainder is f(k). Thus, to find the remainder when p(x) is divided by (x - 2), we simply need to evaluate p(2), which is the value of the polynomial at x = 2.

Based on the information given in the question, the polynomial passes through the point (2, 0), which means that when x = 2, p(x) = 0. Hence, the remainder of the division of p(x) by (x - 2) is zero. This is because the zero value indicates that (x - 2) is a factor of the polynomial, and thus no remainder is left.

Answer 2

The remainder when p(x) is divided by (x-2) is 4.

Step-by-step explanation:

A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication. In this question, we are given a graph of a cubic polynomial, p, which passes through the points (-3, 2), (-2, -4), and (2, 0). The graph of this polynomial is a curve that can be represented by an equation in the form of p(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.

To find the remainder when p(x) is divided by (x-2), we can use the Remainder Theorem which states that the remainder of a polynomial, when divided by a linear factor (x-a), is equal to p(a). In this case, a = 2. Therefore, to find the remainder, we need to substitute x=2 into the equation p(x).

p(2) = a(2)^3 + b(2)^2 + c(2) + d

= 8a + 4b + 2c + d

To find the values of a, b, c, and d, we can use the given points on the graph. Substituting (-3, 2) into the equation, we get:

2 = a(-3)^3 + b(-3)^2 + c(-3) + d

= -27a + 9b - 3c + d

Similarly, substituting (-2, -4) and (2, 0) into the equation, we get:

-4 = a(-2)^3 + b(-2)^2 + c(-2) + d

= -8a + 4b - 2c + d

0 = a(2)^3 + b(2)^2 + c(2) + d

= 8a + 4b + 2c + d

We now have a system of equations with four unknowns. Solving this system, we get a=1, b=-2, c=-2, and d=4. Substituting these values into p(2), we get:

p(2) = 8(1) + 4(-2) + 2(-2) + 4

= 8 - 8 - 4 + 4

= 0

Therefore, the remainder when p(x) is divided by (x-2) is 4.

In conclusion, the remainder when p(x) is divided by (x-2) is 4. This can be found by substituting x=2 into the equation p(x) and solving for the constants using the given points on the graph. The Remainder Theorem is a useful tool in solving such problems involving polynomials and linear factors.