Polynomial Division & Theorems

When working with polynomials, long division helps us break down complex expressions into simpler parts. Just like regular division, we divide term by term and subtract to find quotients and remainders.

The Factor Theorem tells us that if P(c) = 0, then $x-c$ is a factor of polynomial P(x). For example, in the polynomial x⁴-x³+2x-12, we can check if x-2 is a factor by evaluating P(2). Since P(2) = 0, x-2 is indeed a factor!

The Remainder Theorem states that when dividing P(x) by $x-c$, the remainder equals P(c). This gives us a shortcut for finding remainders without doing full polynomial division.

💡 When you're checking if $x-c$ is a factor of a polynomial, just substitute x=c into the polynomial. If you get zero, it's a factor!

Writing Polynomials as Products

When you know a zero of a polynomial, you can use it to write the polynomial as a product of factors. This makes finding all zeros much easier!

For example, if g(x) = x³+2x²-7x-2 and you know 2 is a zero, you can use polynomial division to factor it as: g(x) = $x²+4x+1$$x-2$

To find the remaining zeros, use the quadratic formula on the quadratic factor. For the example above, the quadratic formula gives us x = -2±√3, meaning the polynomial can be written as: g(x) = $x-(-2+√3)$$x-(-2-√3)$$x-2$

This technique helps you break down higher-degree polynomials into manageable pieces, making them easier to solve and understand.

Analyzing Polynomial Graphs

The shape of a polynomial graph tells you a lot about its behavior. A function is increasing when its graph rises from left to right and decreasing when it falls.

You can identify intervals where the function behaves differently. For example, a function might be decreasing on (-∞,1) and (3,7), while increasing on (-1,3) and (7,∞).

An important relationship exists between a polynomial's degree and its local extrema (peaks and valleys). For a polynomial function of degree n>0, the degree is at least 1 more than the number of local extrema. So if a graph shows 3 local extrema, the polynomial must have a degree of at least 4.

🔍 Count the number of "hills" and "valleys" in a polynomial graph, add 1, and you'll know the minimum possible degree of the polynomial!

Direct Variation

Direct variation is when two quantities change proportionally - as one increases, the other increases at the same rate. The equation for direct variation is y = kx, where k is the constant of variation.

In real-world applications, direct variation appears frequently. For instance, when a force acts on an object, the force varies directly with the object's acceleration $F = kA$. If 8 Newtons of force causes 2 m/s² acceleration, then 20 Newtons will cause 5 m/s² acceleration.

To create a direct variation equation from a single point, find the constant k first. If y = 2 when x = 10, then k = 2/10 = 1/5, giving the equation y = (1/5)x. You can then use this equation to find y for any value of x.

🚀 Direct variation is like a see-saw that stays balanced - when one value doubles, the other doubles too!

Inverse Variation

In inverse variation, as one quantity increases, the other decreases proportionally. The equation is y = k/x, where k is the constant of variation.

This relationship appears in many physical situations. For example, when a constant force acts on objects of different masses, the acceleration varies inversely with mass. If an object with 4kg mass accelerates at 19 m/s², we can find that another object with 38kg mass will accelerate at only 2 m/s².

Rational functions often model inverse relationships. When graphing rational functions, identify key features including:

• x-intercepts $where the function crosses the x-axis$
• y-intercepts $where the function crosses the y-axis$
• vertical asymptotes (where the denominator equals zero)
• horizontal asymptotes (the value the function approaches as x gets very large)

⚖️ Think of inverse variation as a seesaw effect: as one value gets 10 times bigger, the other gets 10 times smaller!

Writing Variation Equations

Variation equations describe how quantities relate to each other mathematically. Knowing how to write these equations helps you model real-world relationships.

For inverse variation $y = k/x$, find the constant k using a known point. If y = 8 when x = 3, then 8 = k/3, so k = 24, giving the equation y = 24/x. You can then use this to find y for any x value.

There are three main types of variation equations:

• Direct Variation: x = Ky (x is directly proportional to y)
• Inverse Variation: x = k/y (x is inversely proportional to y)
• Joint Variation: x = Kyz (x varies jointly with y and z)

Understanding these patterns helps you recognize and model relationships in science, economics, and everyday life situations.

Combined Variation & Complex Zeros

Combined variation involves both direct and inverse relationships. For example, the volume (V) of a gas varies directly with temperature (T) and inversely with pressure (P), giving V = KT/P.

To solve problems with combined variation:

1. Write the equation with the constant K
2. Substitute known values to find K
3. Use the complete equation to find unknown values

When working with complex zeros of polynomials, remember that complex zeros always come in conjugate pairs $a+bi and a-bi$. Multiplying complex conjugate expressions like $x-(5-6i)$$x-(5+6i)$ simplifies to x²-10x+61.

The conjugate zeros theorem helps find all zeros of a polynomial. If you know one complex zero $like 1+3i$, you know its conjugate $1-3i$ is also a zero. You can then use polynomial division to find remaining zeros.

🧩 When a polynomial has real coefficients, complex zeros always come in pairs. If a+bi is a zero, then a-bi must also be a zero!

Polynomial Inequalities & Rational Function Asymptotes

Solving polynomial inequalities involves finding where expressions like $x-1$$x+4$$x-5$ ≥ 0 are true. To solve:

1. Find where the expression equals zero $x = 1, -4, 5$
2. These points divide the number line into regions
3. Test each region to determine where the inequality is satisfied

For this example, the solution is $-4,1$ ∪ [5,∞).

Rational functions have important asymptotic behavior:

• Vertical asymptotes occur at zeros of the denominator
• Horizontal asymptotes depend on comparing degrees:
  • If degree of numerator < degree of denominator: y = 0
  • If degrees are equal: y = ratio of leading coefficients
  • If degree of numerator > degree of denominator: no horizontal asymptote

📊 Think of asymptotes as "boundary lines" that the graph approaches but never crosses. They help you quickly sketch the overall shape of rational functions!

Advanced Asymptotes in Rational Functions

Rational functions can have different types of asymptotes depending on the degrees of their numerator and denominator.

For a rational function like f(x) = $x²+2x-3$/$x²-4$, start by finding the vertical asymptotes where the denominator equals zero. After factoring, we get vertical asymptotes at x = -2 and x = 2.

To find the horizontal asymptote, compare the degrees. When the numerator and denominator have the same degree (like x² in this example), the horizontal asymptote is y = ratio of leading coefficients = 1/1 = 1.

A slant asymptote occurs when the degree of the numerator is exactly one more than the denominator. For f(x) = $x²-x+5$/$x+1$, divide to get f(x) = $x-2$+7/$x+1$, giving a slant asymptote of y = x-2.

🔍 When graphing rational functions, identifying the asymptotes first gives you the "skeleton" of the graph, making it much easier to sketch accurately!

Rational Functions with Holes & Descartes' Rule

Rational functions can have holes in their graphs when a factor cancels out. For example, in f(x) = -2x$x-3$/$(x-1)(x-3)$, the $x-3$ factor cancels, creating a hole at x = 3.

To find holes:

1. Factor completely
2. Identify common factors in numerator and denominator
3. Calculate the y-value at this x-value after simplifying

For our example, after simplification, f(x) = -2x/$x-1$, so the hole appears at (3, -3).

Descartes' Rule of Signs helps count possible real zeros of polynomials:

• The number of positive real zeros equals the number of sign variations in P(x) or less by an even number
• The number of negative real zeros equals the sign variations in P$-x$ or less by an even number

For f(x) = -x⁴+3x³-9x²+4x+9, with 3 sign variations, we'll have either 3 or 1 positive real zeros.

🧮 Descartes' Rule won't tell you exactly how many real zeros a polynomial has, but it narrows down the possibilities significantly!