Question

Find a polynomial with integer coefficients that satisfies the given conditions.

R has degree 4 and zeros 1 − 3i and 4, with 4 a zero of multiplicity 2.

R(x) =

Answer 1

`\sf R(x) =\boxed{\sf \;\; x^4 - 10x^3 + 42x^2 - 112x + 160\;\;}`

Explanation:

To find a polynomial with integer coefficients that satisfies the given conditions, we use the factored form of a polynomial.

If
`\sf a` is a root of a polynomial with multiplicity
`\sf m`, then the factor corresponding to
`\sf a` is
`\sf (x - a)^m`.

Given that
`\sf R` has degree 4 and zeros
`\sf 1 - 3i` and
`\sf 4` with
`\sf 4` having multiplicity 2, we can write the factors:

`\sf R(x) = A(x - (1 - 3i))(x - (1 + 3i))(x - 4)^2`

where
`\sf A` is a constant.

Now, let's expand this expression:

`\sf R(x) = A(x - 1 + 3i)(x - 1 - 3i)(x - 4)^2`

To make the coefficients integers, we should consider the conjugates for the complex roots:

`\sf R(x) = A[(x - 1)^2 - (3i)^2](x - 4)^2`

`\sf R(x) = A[(x - 1)^2 + 9](x - 4)^2`

Now, let's expand further:

`\sf R(x) = A[(x^2 - 2x + 1) + 9](x - 4)^2`

`\sf R(x) = A(x^2 - 2x + 10)(x - 4)^2`

Now, expand the squared term:

`\sf R(x) = A(x^2 - 2x + 10)(x^2 - 8x + 16)`

Now, multiply the factors:

`\sf R(x) = A(x^4 - 10x^3 + 42x^2 - 112x + 160)`

Now, compare this expression with the desired answer:

`\sf R(x) = x^4 - 10x^3 + 42x^2 - 112x + 160`

So, the constant
`\sf A` is 1, and the polynomial is indeed:

`\sf R(x) =\boxed{\sf \;\; x^4 - 10x^3 + 42x^2 - 112x + 160\;\;}`

Answer 2

`R(x)=x^4-10x^3+42x^2-112x+160`

Explanation:

Given conditions:

• Polynomial function R(x) with real integer coefficients.
• Degree: 4
• Zeros: (1 - 3i) and 4 (where the zero 4 has a multiplicity of 2).

According to the complex conjugate root theorem, if a polynomial with real coefficients has a complex root (a + bi), then its conjugate (a - bi) is also a root of the polynomial.

Therefore, given that (1 - 3i) is root (zero) of function R(x), then this means that its complex conjugate (1 + 3i) is also a root.

The multiplicity of a zero refers to the number of times the associated factor appears in the factored form of the equation of a polynomial.

As the zero 4 has a multiplicity of 2, then the associated factor appears twice and therefore has an exponent of 2.

Therefore, the factored form of the polynomial is:

`R(x)=(x-4)^2(x-(1-3i))(x-(1+3i))`

`R(x)=(x-4)^2(x-1+3i)(x-1-3i)`

Expand the brackets containing the complex zeros, remembering that i² = -1:

`R(x)=(x-4)^2(x^2-x-3ix-x+1+3i+3ix-3i-9i^2)`

`R(x)=(x-4)^2(x^2-x-x+3ix-3ix+3i-3i+1-9(-1))`

`R(x)=(x-4)^2(x^2-2x+1+9)`

`R(x)=(x-4)^2(x^2-2x+10)`

Expand (x - 4)²:

`R(x)=(x^2-8x+16)(x^2-2x+10)`

Now, expand again so that the polynomial is in standard form:

`R(x)=x^4-2x^3+10x^2-8x^3+16x^2-80x+16x^2-32x+160`

`R(x)=x^4-8x^3-2x^3+10x^2+16x^2+16x^2-80x-32x+160`

`R(x)=x^4-10x^3+42x^2-112x+160`

Therefore, the polynomial with integer coefficients that satisfied the given conditions is:

`\large\boxed{\boxed{R(x)=x^4-10x^3+42x^2-112x+160}}`