IDNLearn.com is committed to providing high-quality answers to your questions. Join our Q&A platform to receive prompt and accurate responses from knowledgeable professionals in various fields.
Sagot :
To find a polynomial function of degree 3 with the given zeros [tex]\(-4\)[/tex], [tex]\(8i\)[/tex], and [tex]\(-8i\)[/tex], we can start by using the fact that if [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the roots (or zeros) of the polynomial, then the polynomial can be constructed as:
[tex]\( f(x) = (x-a)(x-b)(x-c) \)[/tex]
Here, the zeros are [tex]\( -4 \)[/tex], [tex]\( 8i \)[/tex], and [tex]\( -8i \)[/tex]. Therefore, the polynomial function is:
[tex]\[ f(x) = (x + 4)(x - 8i)(x + 8i) \][/tex]
Next, we simplify the expression step-by-step.
First, notice that [tex]\( (x - 8i)(x + 8i) \)[/tex] resembles the difference of squares formula, [tex]\( (a - b)(a + b) = a^2 - b^2 \)[/tex]:
[tex]\[ (x - 8i)(x + 8i) = x^2 - (8i)^2 \][/tex]
Since [tex]\( (8i)^2 = 64i^2 \)[/tex] and knowing that [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ (8i)^2 = 64 \cdot (-1) = -64 \][/tex]
Therefore:
[tex]\[ x^2 - (8i)^2 = x^2 - (-64) = x^2 + 64 \][/tex]
This simplifies our polynomial to:
[tex]\[ f(x) = (x + 4)(x^2 + 64) \][/tex]
Now, distribute [tex]\( x + 4 \)[/tex] across [tex]\( x^2 + 64 \)[/tex]:
[tex]\[ f(x) = x(x^2 + 64) + 4(x^2 + 64) \][/tex]
[tex]\[ f(x) = x^3 + 64x + 4x^2 + 4 \cdot 64 \][/tex]
[tex]\[ f(x) = x^3 + 4x^2 + 64x + 256 \][/tex]
Thus, the polynomial function, simplified, is:
[tex]\[ f(x) = x^3 + 4x^2 + 64x + 256 \][/tex]
So, the polynomial function of degree 3 with the given zeros [tex]\(-4\)[/tex], [tex]\(8i\)[/tex], and [tex]\(-8i\)[/tex], and a leading coefficient of 1, is:
[tex]\[ \boxed{x^3 + 4x^2 + 64x + 256} \][/tex]
[tex]\( f(x) = (x-a)(x-b)(x-c) \)[/tex]
Here, the zeros are [tex]\( -4 \)[/tex], [tex]\( 8i \)[/tex], and [tex]\( -8i \)[/tex]. Therefore, the polynomial function is:
[tex]\[ f(x) = (x + 4)(x - 8i)(x + 8i) \][/tex]
Next, we simplify the expression step-by-step.
First, notice that [tex]\( (x - 8i)(x + 8i) \)[/tex] resembles the difference of squares formula, [tex]\( (a - b)(a + b) = a^2 - b^2 \)[/tex]:
[tex]\[ (x - 8i)(x + 8i) = x^2 - (8i)^2 \][/tex]
Since [tex]\( (8i)^2 = 64i^2 \)[/tex] and knowing that [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ (8i)^2 = 64 \cdot (-1) = -64 \][/tex]
Therefore:
[tex]\[ x^2 - (8i)^2 = x^2 - (-64) = x^2 + 64 \][/tex]
This simplifies our polynomial to:
[tex]\[ f(x) = (x + 4)(x^2 + 64) \][/tex]
Now, distribute [tex]\( x + 4 \)[/tex] across [tex]\( x^2 + 64 \)[/tex]:
[tex]\[ f(x) = x(x^2 + 64) + 4(x^2 + 64) \][/tex]
[tex]\[ f(x) = x^3 + 64x + 4x^2 + 4 \cdot 64 \][/tex]
[tex]\[ f(x) = x^3 + 4x^2 + 64x + 256 \][/tex]
Thus, the polynomial function, simplified, is:
[tex]\[ f(x) = x^3 + 4x^2 + 64x + 256 \][/tex]
So, the polynomial function of degree 3 with the given zeros [tex]\(-4\)[/tex], [tex]\(8i\)[/tex], and [tex]\(-8i\)[/tex], and a leading coefficient of 1, is:
[tex]\[ \boxed{x^3 + 4x^2 + 64x + 256} \][/tex]
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for trusting IDNLearn.com. We’re dedicated to providing accurate answers, so visit us again for more solutions.