The polynomial function p is defined by p(x) = x^4 - 8x^3 + 20x^2 - 16x + c, where c is a … — SAT

The polynomial function $p$ is defined by $p(x) = x^4 - 8x^3 + 20x^2 - 16x + c$ , where $c$ is a constant. In the $xy$ -plane, the graph of $y = p(x)$ is tangent to the $x$ -axis at two distinct points. What is the value of $c$ ?

Answer: 4

Answer

The correct answer is 4. A quartic polynomial that is tangent to the x-axis at two distinct points must have two roots of multiplicity 2. Thus, it can be factored as p(x) = (x - r_1)^2(x - r_2)^2 = [x^2 - (r_1 + r_2)x + r_1 r_2]^2. Setting S = r_1 + r_2 and P = r_1 r_2, we expand this to p(x) = x^4 - 2Sx^3 + (S^2 + 2P)x^2 - 2SPx + P^2. Matching this with the given equation x^4 - 8x^3 + 20x^2 - 16x + c shows that -2S = -8, so S = 4. Using the quadratic term, S^2 + 2P = 20, which yields 16 + 2P = 20, so P = 2. Finally, the constant term is c = P^2 = 2^2 = 4.

Step-by-Step Solution

Express the polynomial using its roots and multiplicities.

p(x) = (x - r_1)^2(x - r_2)^2

Since the graph of the quartic polynomial is tangent to the x-axis at two distinct points, it must have two real roots, each with a multiplicity of 2.

Group the terms and introduce variables for the sum and product of the roots.

p(x) = [x^2 - Sx + P]^2, where S = r_1 + r_2 and P = r_1 r_2

This simplifies expansion by using the symmetric properties of roots.

Expand the squared quadratic expression.

p(x) = x^4 - 2Sx^3 + (S^2 + 2P)x^2 - 2SPx + P^2

To compare coefficients with the standard form of the given polynomial.

Compare the coefficients of the x^3 term.

-2S = -8, which gives S = 4

To determine the sum of the roots from the given cubic coefficient.

Compare the coefficients of the x^2 term using the value of S.

S^2 + 2P = 20, which becomes 16 + 2P = 20, leading to P = 2

To find the product of the roots using the quadratic coefficient.

Verify the linear coefficient using S and P.

-2SP = -2(4)(2) = -16

To ensure consistency with the given coefficient of x.

Determine the constant term c.

c = P^2 = 2^2 = 4

To find the final value of the constant c.

Key Concept

Representing a polynomial function in factored form based on the multiplicity of its roots and using coefficient matching to determine unknown constants.

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