11.3 Polynomials (Polinomlar), Quadratic Equations, Vieta's Formulas & Parabolas

Polynomials (Polinomlar), Quadratic Equations, Vieta's Formulas & Parabolas

Polynomials, quadratic equations (ikinci dereceden denklemler), and parabolas (parabol) form a tightly connected triad on the TR-YÖS. Remainder problems turn into one-line evaluations via the Remainder Theorem, root expressions simplify without ever using the quadratic formula via Vieta's relations, and parabola geometry reduces to vertex symmetry and discriminant tangency.

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1. Polynomials (Polinomlar): Exponents, Degrees & Coefficient Rules

An algebraic expression $P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ ($a_k \in \mathbb{R}$) is a polynomial if and only if every exponent of $x$ with a nonzero coefficient belongs to the natural numbers $\mathbb{N} = \{0, 1, 2, 3, \dots\}$.

• Exam Trap: If $P(x) = 3x^{n-4} + 2x^{\frac{12}{n}} - 5$ is a polynomial, you must enforce both $n - 4 \ge 0$ and $\frac{12}{n} \in \mathbb{N}$, which restricts $n$ to divisors of $12$ that are at least $4$: $n \in \{4, 6, 12\}$.

Degree Rules (Derece Kuralları)

Let $\deg[P(x)] = m$ and $\deg[Q(x)] = n$ with $m > n$:

• $\deg[P(x) \pm Q(x)] = m$ (if $m = n$, $\deg[P \pm Q] \le m$ because leading terms may cancel)
• $\deg[P(x) \cdot Q(x)] = m + n$
• $\deg\left[\frac{P(x)}{Q(x)}\right] = m - n$ (when $Q(x)$ divides $P(x)$)
• $\deg[P^k(x^r)] = k \cdot r \cdot m$ and $\deg[P(Q(x))] = m \cdot n$

Coefficient Sums (Katsayılar Toplamı) & Constant Term (Sabit Terim)

For any polynomial expression (whether $P(x)$, $P(2x - 3)$, or $Q(x + 1)$):

1. Sum of Coefficients (Katsayılar Toplamı): Substitute $x = 1$ into the polynomial explicitly named in the question! For example, the sum of the coefficients of $P(2x - 3)$ is $P(2(1) - 3) = P(-1)$.
2. Constant Term (Sabit Terim): Substitute $x = 0$ into the polynomial explicitly named in the question! For example, the constant term of $P(2x - 3)$ is $P(2(0) - 3) = P(-3)$.
3. Even- and Odd-Degree Coefficient Sums: For $P(x)$ itself:

$$\text{Even-degree coefficients} = \frac{P(1) + P(-1)}{2}, \qquad \text{Odd-degree coefficients} = \frac{P(1) - P(-1)}{2}$$

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2. Polynomial Division & the Remainder Theorem (Kalan Bulma)

In polynomial division $P(x) = D(x) \cdot Q(x) + R(x)$, the remainder $R(x)$ is either $0$ or satisfies $\deg[R(x)] < \deg[D(x)]$. You never need long division on the TR-YÖS if you set the divisor equal to zero:

1. Linear Divisor $ax + b$: Set $ax + b = 0 \implies x = -\frac{b}{a}$. The remainder is the constant $R = P\left(-\frac{b}{a}\right)$. Specifically, $P(x)$ is divisible by $x - a$ (tam bölünür) if and only if $P(a) = 0$.
2. Factorable Quadratic Divisor $(x - r_1)(x - r_2)$: Since the divisor has degree $2$, the remainder has at most degree $1$: write $R(x) = mx + n$. Substitute the two roots of the divisor:

$$P(r_1) = m r_1 + n \quad \text{and} \quad P(r_2) = m r_2 + n$$

and solve the $2 \times 2$ linear system for $m$ and $n$. 3. Power Divisor $x^n - c$: Set $x^n - c = 0 \implies x^n = c$, and replace every power of $x^n$ in $P(x)$ with $c$!

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3. Quadratic Equations & Vieta's Formulas (Kök-Katsayı İlişkileri)

For the quadratic equation $ax^2 + bx + c = 0$ ($a \ne 0$), the discriminant is $\Delta = b^2 - 4ac$:

• $\Delta > 0$: Two distinct real roots $x_{1,2} = \frac{-b \pm \sqrt{\Delta}}{2a}$.
• $\Delta = 0$: One double/coincident real root (çakışık / çift katlı kök) $x_1 = x_2 = -\frac{b}{2a}$; $ax^2 + bx + c$ is a scalar multiple of a squared linear factor.
• $\Delta < 0$: No real roots (reel kök yoktur); two complex conjugate roots.

Vieta's Formulas (Kökler Toplamı, Çarpımı ve Farkı)

Never solve for $x_1$ and $x_2$ individually when asked for a symmetric combination of the roots:

$$T = x_1 + x_2 = -\frac{b}{a}, \qquad Ç = x_1 \cdot x_2 = \frac{c}{a}, \qquad \lvert x_1 - x_2 \rvert = \frac{\sqrt{\Delta}}{\lvert a \rvert}$$

• Essential Symmetric Identities:
  • $\frac{1}{x_1} + \frac{1}{x_2} = \frac{x_1 + x_2}{x_1 x_2} = -\frac{b}{c}$
  • $x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2 = T^2 - 2Ç$
  • $x_1^3 + x_2^3 = (x_1 + x_2)^3 - 3x_1 x_2(x_1 + x_2) = T^3 - 3TÇ$
• Constructing a Quadratic from Its Roots: Any quadratic equation whose roots have sum $T = x_1 + x_2$ and product $Ç = x_1 x_2$ is written directly as:

$$x^2 - Tx + Ç = 0$$

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4. Parabolas (Parabol — $y = ax^2 + bx + c$)

The graph of $f(x) = ax^2 + bx + c$ ($a \ne 0$) is a parabola opening upward (kollar yukarı, minimum at vertex) if $a > 0$ and downward (kollar aşağı, maximum at vertex) if $a < 0$.

Vertex (Tepe Noktası $T(r, k)$) & Axis of Symmetry

$$r = -\frac{b}{2a} = \frac{x_1 + x_2}{2}, \qquad k = f(r) = \frac{4ac - b^2}{4a} = -\frac{\Delta}{4a}$$

The vertical line $x = r$ is the axis of symmetry (simetri ekseni): any two points on the parabola with the same $y$-value, $f(p) = f(q)$ ($p \ne q$), satisfy $\frac{p + q}{2} = r$.

Three Forms of a Parabola Equation

1. Standard Form: $y = ax^2 + bx + c$ (use when given $3$ arbitrary points including the $y$-intercept $(0, c)$).
2. Vertex Form (Tepe Noktası Bilinen Parabol): Given the vertex $T(r, k)$ and one additional point:

$$y = a(x - r)^2 + k$$

1. Factored / Root Form (Eksenleri Kestiği Noktalar Bilinen Parabol): Given the two $x$-intercepts $(x_1, 0)$ and $(x_2, 0)$:

$$y = a(x - x_1)(x - x_2)$$

Line-Parabola Intersection

To find how the parabola $y = ax^2 + bx + c$ and the line $y = mx + n$ intersect, equate them to form the common equation (ortak denklem) $ax^2 + (b - m)x + (c - n) = 0$ and compute its discriminant $\Delta_{\text{common}}$:

• $\Delta_{\text{common}} > 0$: Line intersects the parabola at $2$ distinct points.
• $\Delta_{\text{common}} = 0$: Line is tangent (teğet) to the parabola at $1$ point.
• $\Delta_{\text{common}} < 0$: Line and parabola do not intersect.

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Summary Reference Table

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Worked Examples

Worked Example 1: Coefficient Sum, Constant Term & Quadratic Divisor Remainder

Problem: For a polynomial $P(x)$, the sum of the coefficients of $P(x - 1)$ is $7$, and the constant term of $P(2x + 3)$ is $-5$. Find the remainder $R(x)$ when $P(x)$ is divided by $x^2 - 3x$.

Solution:

1. "Sum of the coefficients of $P(x - 1) = 7$" means set $x = 1$: $P(1 - 1) = P(0) = 7$.
2. "Constant term of $P(2x + 3) = -5$" means set $x = 0$: $P(2(0) + 3) = P(3) = -5$.
3. Since the divisor $x^2 - 3x = x(x - 3)$ has degree $2$ with roots $x = 0$ and $x = 3$, write the remainder as $R(x) = mx + n$, so $P(x) = x(x - 3)Q(x) + mx + n$.
4. Substitute $x = 0$: $P(0) = n = 7$. Substitute $x = 3$: $P(3) = 3m + 7 = -5 \implies 3m = -12 \implies m = -4$. Thus $R(x) = \mathbf{-4x + 7}$.

Worked Example 2: Vieta's Symmetric Sums & Parabola Vertex/Tangency

Problem:

1. If $x_1$ and $x_2$ are the roots of $2x^2 - 6x - 5 = 0$, find $x_1^2 + x_2^2$ and $\frac{1}{x_1} + \frac{1}{x_2}$.
2. A parabola $y = f(x)$ has vertex $T(2, -9)$ and passes through $(0, -5)$. Find the value of $m$ for which the line $y = 4x + m$ is tangent to the parabola.

Solution:

1. By Vieta's formulas, $x_1 + x_2 = -\frac{-6}{2} = 3$ and $x_1 x_2 = -\frac{5}{2}$. Thus:

$$x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2 = 3^2 - 2\left(-\frac{5}{2}\right) = 9 + 5 = \mathbf{14}, \qquad \frac{1}{x_1} + \frac{1}{x_2} = \frac{x_1 + x_2}{x_1 x_2} = \frac{3}{-5/2} = \mathbf{-\frac{6}{5}}$$

1. Using vertex form $y = a(x - 2)^2 - 9$ and substituting $(0, -5)$ gives $-5 = 4a - 9 \implies a = 1$, so $y = (x - 2)^2 - 9 = x^2 - 4x - 5$. Setting the parabola equal to $y = 4x + m$ yields the common equation $x^2 - 8x - (5 + m) = 0$. For tangency, set $\Delta_{\text{common}} = 0$:

$$(-8)^2 - 4(1)(-(5 + m)) = 64 + 20 + 4m = 0 \implies 4m = -84 \implies m = \mathbf{-21}$$

Degree rules apply to nonzero polynomials and positive integer powers; the zero polynomial needs separate treatment. A quadratic divisor with distinct roots gives two independent remainder equations. For a repeated root, one evaluation is insufficient and another condition such as a derivative value is needed. Root-difference formulas above assume real roots ($\Delta\ge0$), and reciprocal-root expressions require $c\ne0$.

Review the Method

A parabola y = ax^2 + bx + c has vertex T(r, k) with r = -b/(2a) = (x_1 + x_2)/2 and k = f(r) = (4ac - b^2)/(4a), and can be written in standard form, vertex form y = a(x - r)^2 + k, or factored root form y = a(x - x_1)(x - x_2).