Question

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.

Answer 1

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