Find the function P defined by a polynomial of degree 3 with real coefficients that satisfy the given conditions. The zeros are -3,-1, and 4. The leading coefficient is -4.

Question

asked Dec 28, 2021 158k views

1 Answer

Derrick Beining · Answer 1 · 2022-01-03T06:50:31+0000

Given:

Degree of polynomial = 3

Zeros are -3,-1, and 4.

The leading coefficient is -4.

To find:

The polynomial.

Solution:

The general form of a polynomial is

P(x)=a(x-c_1)^(m_1)(x-c_2)^(m_2)...(x-c_n)^(m_n)

where, a is a constant,
c_1,c_2,...,c_n are zeros with multiplicity
m_1,m_2,...,m_n respectively.

Zeros of the polynomial are -3,-1, and 4. So,

P(x)=a(x-(-3))(x-(-1))(x-4)

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

P(x)=a(x^2+3x+x+3)(x-4)

P(x)=a(x^2+4x+3)(x-4)

P(x)=a(x^3+4x^2+3x-4x^2-16x-12)

P(x)=a(x^3-13x-12)

P(x)=ax^3-13ax-12a

Here, leading coefficient is a.

The leading coefficient is -4. So, a=-4.

P(x)=(-4)x^3-13(-4)x-12(-4)

P(x)=-4x^3+52x+48

Therefore, the required polynomial is
.