Answer :
To factor the polynomial [tex]\( P(x) = 3x^5 - 16x^4 + 35x^3 - 70x^2 + 92x - 24 \)[/tex] given that [tex]\( 2i \)[/tex] is a zero, we follow these steps:
1. Identify the Known Zeros: Since [tex]\( 2i \)[/tex] is a zero, its complex conjugate, [tex]\(-2i\)[/tex], must also be a zero because polynomial coefficients are real numbers.
2. Form the Quadratic Factor: The zeros [tex]\( 2i \)[/tex] and [tex]\(-2i\)[/tex] imply that [tex]\((x - 2i)(x + 2i) = x^2 + 4\)[/tex] is a factor of the polynomial [tex]\( P(x) \)[/tex].
3. Divide the Polynomial by the Quadratic Factor: We need to divide [tex]\( P(x) \)[/tex] by the quadratic factor [tex]\( x^2 + 4 \)[/tex] to reduce the polynomial:
Dividing, we get:
[tex]\[
\frac{P(x)}{x^2 + 4} = 3x^3 - 16x^2 + 35x - 70
\][/tex]
4. Factor the Resulting Polynomial: Next, we need to factor the resulting cubic polynomial [tex]\( 3x^3 - 16x^2 + 35x - 70 \)[/tex].
5. Find the Rational Roots: Testing small rational numbers, we find that [tex]\( x = 2 \)[/tex] is a root. This means [tex]\( x - 2 \)[/tex] is a factor.
6. Factor the Cubic Polynomial: Synthetic or polynomial division of [tex]\( 3x^3 - 16x^2 + 35x - 70 \)[/tex] by [tex]\( x - 2 \)[/tex] gives:
[tex]\[
3x^2 - 10x + 35
\][/tex]
7. Factor the Quadratic Polynomial: The quadratic [tex]\( 3x^2 - 10x + 35 \)[/tex] can be further broken down into two linear factors [tex]\( (x - 3) \)[/tex] and [tex]\( (3x - 1) \)[/tex].
8. Write the Full Factored Form: Including all identified factors, the polynomial can be expressed as:
[tex]\[
P(x) = (x - 2i)(x + 2i)(x - 2)(x - 3)(3x - 1)
\][/tex]
Thus, the fully factored form of [tex]\( P(x) \)[/tex] is:
[tex]\[
P(x) = (x^2 + 4)(x - 2)(x - 3)(3x - 1)
\][/tex]
1. Identify the Known Zeros: Since [tex]\( 2i \)[/tex] is a zero, its complex conjugate, [tex]\(-2i\)[/tex], must also be a zero because polynomial coefficients are real numbers.
2. Form the Quadratic Factor: The zeros [tex]\( 2i \)[/tex] and [tex]\(-2i\)[/tex] imply that [tex]\((x - 2i)(x + 2i) = x^2 + 4\)[/tex] is a factor of the polynomial [tex]\( P(x) \)[/tex].
3. Divide the Polynomial by the Quadratic Factor: We need to divide [tex]\( P(x) \)[/tex] by the quadratic factor [tex]\( x^2 + 4 \)[/tex] to reduce the polynomial:
Dividing, we get:
[tex]\[
\frac{P(x)}{x^2 + 4} = 3x^3 - 16x^2 + 35x - 70
\][/tex]
4. Factor the Resulting Polynomial: Next, we need to factor the resulting cubic polynomial [tex]\( 3x^3 - 16x^2 + 35x - 70 \)[/tex].
5. Find the Rational Roots: Testing small rational numbers, we find that [tex]\( x = 2 \)[/tex] is a root. This means [tex]\( x - 2 \)[/tex] is a factor.
6. Factor the Cubic Polynomial: Synthetic or polynomial division of [tex]\( 3x^3 - 16x^2 + 35x - 70 \)[/tex] by [tex]\( x - 2 \)[/tex] gives:
[tex]\[
3x^2 - 10x + 35
\][/tex]
7. Factor the Quadratic Polynomial: The quadratic [tex]\( 3x^2 - 10x + 35 \)[/tex] can be further broken down into two linear factors [tex]\( (x - 3) \)[/tex] and [tex]\( (3x - 1) \)[/tex].
8. Write the Full Factored Form: Including all identified factors, the polynomial can be expressed as:
[tex]\[
P(x) = (x - 2i)(x + 2i)(x - 2)(x - 3)(3x - 1)
\][/tex]
Thus, the fully factored form of [tex]\( P(x) \)[/tex] is:
[tex]\[
P(x) = (x^2 + 4)(x - 2)(x - 3)(3x - 1)
\][/tex]
