Problems 89-92 are based on material learned earlier in the...

Problems 89-92 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
$$
\text { Find the complex zeros of } f(x)=x^{4}-32 x^{2}-144
$$

Key Concepts

Polynomial Factorization

This concept involves breaking down a polynomial into simpler factors that multiply together to give the original polynomial. In the context of finding zeros, factoring the polynomial helps to reduce the problem into solving for roots of lower degree polynomials, making the process more manageable.

Substitution Technique

The substitution technique simplifies complex polynomial equations by replacing a repeated expression with a single variable. For example, when a polynomial in x contains terms like x? and x², substituting u = x² can convert it into a quadratic in u, which is easier to solve.

Quadratic Formula

The quadratic formula is a fundamental tool for solving quadratic equations. Once a higher-degree polynomial is reduced to a quadratic form through substitution, the quadratic formula gives the solutions (roots) for the substituted variable, which can then be translated back to the original variable.

Complex Numbers

Complex numbers extend the real numbers and include an imaginary unit that satisfies i² = -1. When solving polynomial equations, especially those with negative discriminants in the quadratic formula, the solutions can be complex. Understanding complex numbers is essential for fully identifying all zeros of a polynomial.

Transcript

00:02 This problem, we're asked to find the roots of this polynomial function.

00:06 So anytime you see exponents 4 and the highest exponent 4 and then the middle term is, you know, basically half of the highest exponents, you know that you can rewrite this in terms of quadratic equation.

00:22 So let's say, you know, that u equals to x squared.

00:27 Okay, so therefore i can rewrite this equation, as this is the same thing as u squared plus 21 u minus 100.

00:45 Okay, so you can see why this is the same because remember u is x squared so i replace x square with u and then x squared raised to another squared is going to give you x to the fourths.

00:58 So now i'm going to set equal to zero and solve this quadratic equation.

01:03 So i can solve this by factoring.

01:06 So i can think of two numbers that will multiply give me negative 100, but add give me 21.

01:14 And the two number that i can think of immediately is 4 and 25, because 4 times 25 is 100.

01:21 But one of them has to be negative.

01:23 So we want a positive 21, so therefore 4 has to be negative.

01:26 So therefore, i can rewrite this as in the factor form of u minus 4.

01:33 4 times u plus 25.

01:37 So because the set equal to 0 to solve for you, you're going to say each factor equal to 0.

01:43 So u minus 4 equals to 0, u plus 25 equals to 0.

01:48 So therefore, u equals to 4 and u equals to negative 25, okay? but remember, we want to find out what x equals to, not what u equals to...

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