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Answer :
To write the polynomial [tex]\( P(x) = 3x^5 - 4x^4 + 39x^3 - 54x^2 - 144x + 160 \)[/tex] in factored form, follow these steps:
1. Identify the given zero: We are given that [tex]\( 4i \)[/tex] is a zero of the polynomial. Since the coefficients of the polynomial are real numbers, any non-real zeros must occur in conjugate pairs. Therefore, [tex]\(-4i\)[/tex] is also a zero.
2. Factor out corresponding quadratic polynomial: The zeros [tex]\( 4i \)[/tex] and [tex]\(-4i\)[/tex] imply a factor of the form [tex]\((x - 4i)(x + 4i)\)[/tex]. We can simplify this using the identity for the difference of squares:
[tex]\[
(x - 4i)(x + 4i) = x^2 - (4i)^2 = x^2 + 16
\][/tex]
3. Perform polynomial division: Divide the polynomial [tex]\( P(x) \)[/tex] by the factor [tex]\( x^2 + 16 \)[/tex]. This will help find the quotient, which is a polynomial of degree 3.
4. Find remaining zeros and their factors: The quotient from the division in step 3 is a cubic polynomial, say [tex]\( Q(x) \)[/tex]. Factor [tex]\( Q(x) \)[/tex] completely to find the remaining zeros.
In this case, the zeros of [tex]\( Q(x) \)[/tex] are [tex]\( x = 2 \)[/tex], [tex]\( x = 1 \)[/tex], and [tex]\( x = -\frac{5}{3} \)[/tex].
5. Combine all factors: The polynomial [tex]\( P(x) \)[/tex] can now be written in factored form by combining the factors from the roots:
[tex]\[
P(x) = 3(x - 2)(x - 1)\left(x + \frac{5}{3}\right)(x - 4i)(x + 4i)
\][/tex]
So the completely factored form of the polynomial is:
[tex]\[
P(x) = 3(x - 2)(x - 1)\left(x + \frac{5}{3}\right)(x - 4i)(x + 4i)
\][/tex]
1. Identify the given zero: We are given that [tex]\( 4i \)[/tex] is a zero of the polynomial. Since the coefficients of the polynomial are real numbers, any non-real zeros must occur in conjugate pairs. Therefore, [tex]\(-4i\)[/tex] is also a zero.
2. Factor out corresponding quadratic polynomial: The zeros [tex]\( 4i \)[/tex] and [tex]\(-4i\)[/tex] imply a factor of the form [tex]\((x - 4i)(x + 4i)\)[/tex]. We can simplify this using the identity for the difference of squares:
[tex]\[
(x - 4i)(x + 4i) = x^2 - (4i)^2 = x^2 + 16
\][/tex]
3. Perform polynomial division: Divide the polynomial [tex]\( P(x) \)[/tex] by the factor [tex]\( x^2 + 16 \)[/tex]. This will help find the quotient, which is a polynomial of degree 3.
4. Find remaining zeros and their factors: The quotient from the division in step 3 is a cubic polynomial, say [tex]\( Q(x) \)[/tex]. Factor [tex]\( Q(x) \)[/tex] completely to find the remaining zeros.
In this case, the zeros of [tex]\( Q(x) \)[/tex] are [tex]\( x = 2 \)[/tex], [tex]\( x = 1 \)[/tex], and [tex]\( x = -\frac{5}{3} \)[/tex].
5. Combine all factors: The polynomial [tex]\( P(x) \)[/tex] can now be written in factored form by combining the factors from the roots:
[tex]\[
P(x) = 3(x - 2)(x - 1)\left(x + \frac{5}{3}\right)(x - 4i)(x + 4i)
\][/tex]
So the completely factored form of the polynomial is:
[tex]\[
P(x) = 3(x - 2)(x - 1)\left(x + \frac{5}{3}\right)(x - 4i)(x + 4i)
\][/tex]
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Rewritten by : Jeany
