Answer :
To find all the zeros of the function [tex]\( f(x) = x^4 - 21x^2 - 100 \)[/tex], we'll follow a methodical approach.
1. Identify the Type of Polynomial: We have a polynomial of degree 4. The zeros of this polynomial can be real or complex numbers.
2. Substitution to Simplify the Equation: Recognize that the equation is quadratic in form with respect to [tex]\( x^2 \)[/tex]. Let's make a substitution where [tex]\( y = x^2 \)[/tex]. This changes the function from [tex]\( f(x) = x^4 - 21x^2 - 100 \)[/tex] to a quadratic form:
[tex]\[
f(y) = y^2 - 21y - 100
\][/tex]
3. Solve the Quadratic Equation: Use the quadratic formula to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = -21 \)[/tex], and [tex]\( c = -100 \)[/tex].
[tex]\[
y = \frac{-(-21) \pm \sqrt{(-21)^2 - 4 \times 1 \times (-100)}}{2 \times 1}
\][/tex]
[tex]\[
y = \frac{21 \pm \sqrt{441 + 400}}{2}
\][/tex]
[tex]\[
y = \frac{21 \pm \sqrt{841}}{2}
\][/tex]
[tex]\[
y = \frac{21 \pm 29}{2}
\][/tex]
This gives us two solutions for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{50}{2} = 25
\][/tex]
[tex]\[
y = \frac{-8}{2} = -4
\][/tex]
4. Convert Back to [tex]\( x \)[/tex]: Remember that [tex]\( y = x^2 \)[/tex], so now we solve for [tex]\( x \)[/tex].
For [tex]\( y = 25 \)[/tex]:
[tex]\[
x^2 = 25 \implies x = \pm 5
\][/tex]
For [tex]\( y = -4 \)[/tex]:
[tex]\[
x^2 = -4 \implies x = \pm 2i
\][/tex]
5. List All Zeros: The zeros of the polynomial [tex]\( f(x) = x^4 - 21x^2 - 100 \)[/tex] are:
[tex]\[
x = -5, 5, -2i, 2i
\][/tex]
And there you have it, the zeros are [tex]\(-5\)[/tex], [tex]\(5\)[/tex], [tex]\(-2i\)[/tex], and [tex]\(2i\)[/tex]. These include both real numbers and purely imaginary numbers, as expected for a degree 4 polynomial.
1. Identify the Type of Polynomial: We have a polynomial of degree 4. The zeros of this polynomial can be real or complex numbers.
2. Substitution to Simplify the Equation: Recognize that the equation is quadratic in form with respect to [tex]\( x^2 \)[/tex]. Let's make a substitution where [tex]\( y = x^2 \)[/tex]. This changes the function from [tex]\( f(x) = x^4 - 21x^2 - 100 \)[/tex] to a quadratic form:
[tex]\[
f(y) = y^2 - 21y - 100
\][/tex]
3. Solve the Quadratic Equation: Use the quadratic formula to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = -21 \)[/tex], and [tex]\( c = -100 \)[/tex].
[tex]\[
y = \frac{-(-21) \pm \sqrt{(-21)^2 - 4 \times 1 \times (-100)}}{2 \times 1}
\][/tex]
[tex]\[
y = \frac{21 \pm \sqrt{441 + 400}}{2}
\][/tex]
[tex]\[
y = \frac{21 \pm \sqrt{841}}{2}
\][/tex]
[tex]\[
y = \frac{21 \pm 29}{2}
\][/tex]
This gives us two solutions for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{50}{2} = 25
\][/tex]
[tex]\[
y = \frac{-8}{2} = -4
\][/tex]
4. Convert Back to [tex]\( x \)[/tex]: Remember that [tex]\( y = x^2 \)[/tex], so now we solve for [tex]\( x \)[/tex].
For [tex]\( y = 25 \)[/tex]:
[tex]\[
x^2 = 25 \implies x = \pm 5
\][/tex]
For [tex]\( y = -4 \)[/tex]:
[tex]\[
x^2 = -4 \implies x = \pm 2i
\][/tex]
5. List All Zeros: The zeros of the polynomial [tex]\( f(x) = x^4 - 21x^2 - 100 \)[/tex] are:
[tex]\[
x = -5, 5, -2i, 2i
\][/tex]
And there you have it, the zeros are [tex]\(-5\)[/tex], [tex]\(5\)[/tex], [tex]\(-2i\)[/tex], and [tex]\(2i\)[/tex]. These include both real numbers and purely imaginary numbers, as expected for a degree 4 polynomial.
