7.4 Polynomial Equations, Roots, and Graphs

Polynomial Arithmetic, Roots, Radical & Rational Equations

Core Foundation: Advanced algebra on the ACCUPLACER requires fluency with higher-degree polynomial functions, root-finding theorems, algebraic fractions, and radical equations. You will be tested on polynomial division algorithms, the behavior of polynomial graphs at infinity and at roots of varying multiplicity, and the essential skill of identifying extraneous solutions in rational and radical equations.

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Polynomial Arithmetic and Division Techniques

A polynomial function of degree $n$ is an algebraic expression of the form: $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \quad (a_n \neq 0)$

Basic Arithmetic Operations

• Addition and Subtraction: Combine like terms (terms having identical variables raised to identical exponent powers). When subtracting, distribute the negative sign across every term in the subtracted polynomial: $(4x^3 - 2x^2 + 5x) - (x^3 + 3x^2 - 7) = (4 - 1)x^3 + (-2 - 3)x^2 + 5x - (-7) = 3x^3 - 5x^2 + 5x + 7$
• Multiplication: Distribute every term of the first polynomial across every term of the second (using the distributive property or the box grid method) and sum like terms: $(2x - 3)(x^2 + 4x - 5) = 2x(x^2 + 4x - 5) - 3(x^2 + 4x - 5) = 2x^3 + 8x^2 - 10x - 3x^2 - 12x + 15 = 2x^3 + 5x^2 - 22x + 15$

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Polynomial Long Division vs. Synthetic Division

When dividing a polynomial $P(x)$ (dividend) by $D(x)$ (divisor), the division algorithm yields a quotient $Q(x)$ and a remainder $R(x)$:

$\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)} \iff P(x) = D(x) \cdot Q(x) + R(x)$

+-----------------------------------------------------------------------------------------+
|                     POLYNOMIAL DIVISION METHOD COMPARISON                               |
+-----------------------+---------------------------------+-------------------------------+
| Dimension             | Polynomial Long Division        | Synthetic Division            |
+-----------------------+---------------------------------+-------------------------------+
| Applicable Divisors   | Any polynomial divisor D(x)     | Strictly linear monic divisors|
|                       | (Linear, Quadratic, etc.)       | of the form (x - c)           |
+-----------------------+---------------------------------+-------------------------------+
| Variable Handling     | Tracks all variable powers      | Uses numerical coefficients   |
|                       | explicitly                      | only (compact shorthand)      |
+-----------------------+---------------------------------+-------------------------------+
| Missing Powers Rule   | Must insert 0x^k placeholders   | Must insert 0 placeholders in |
|                       | for missing terms               | coefficient array             |
+-----------------------+---------------------------------+-------------------------------+
| Arithmetic Mechanics  | Divide, Multiply, Subtract,     | Bring down, Multiply by c,    |
|                       | Bring down next term            | Add column                    |
+-----------------------+---------------------------------+-------------------------------+

Synthetic Division Step-by-Step

To divide $P(x) = 2x^3 - 3x^2 - 11x + 6$ by $(x - 3)$:

1. Set divisor $x - 3 = 0 \implies c = 3$.
2. Write the coefficients of $P(x)$: $[2, , -3, , -11, , 6]$.
3. Bring down the leading coefficient $2$.
4. Multiply $2 \times 3 = 6$; add to $-3$ to get $3$.
5. Multiply $3 \times 3 = 9$; add to $-11$ to get $-2$.
6. Multiply $-2 \times 3 = -6$; add to $6$ to get $0$ (Remainder = $0$).
7. The resulting quotient polynomial is $Q(x) = 2x^2 + 3x - 2$.

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The Remainder & Factor Theorems

These two fundamental algebraic theorems allow you to evaluate polynomials and determine roots rapidly without performing full polynomial division.

The Polynomial Remainder Theorem

Theorem: If a polynomial $P(x)$ is divided by a linear factor $(x - c)$, the resulting constant remainder is equal to the direct function evaluation $P(c)$: $\text{Remainder } R = P(c)$

Example: To find the remainder when $P(x) = x^3 - 4x^2 + 7x - 5$ is divided by $(x - 2)$, simply compute $P(2)$: $P(2) = (2)^3 - 4(2)^2 + 7(2) - 5 = 8 - 16 + 14 - 5 = 1$ The remainder is $1$.

The Factor Theorem

Theorem: A linear expression $(x - c)$ is an exact algebraic factor of a polynomial $P(x)$ if and only if $P(c) = 0$ (meaning $c$ is a root / zero of $P(x)$).

If synthetic division or direct substitution yields $P(c) = 0$, then $(x - c)$ divides evenly into $P(x)$ with zero remainder, and $x = c$ is an $x$-intercept of the graph.

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Polynomial Graph End Behavior, Turning Points & Multiplicity

The qualitative shape of any polynomial graph is governed by its degree and roots.

The Leading Coefficient Test (End Behavior)

As $x \to \pm \infty$, the highest-degree term $a_n x^n$ dominates all other terms:

+-----------------------------------------------------------------------------------------+
|                        POLYNOMIAL END BEHAVIOR MATRIX                                   |
+-----------------------+---------------------------------+-------------------------------+
| Degree Type           | Leading Coefficient a_n > 0     | Leading Coefficient a_n < 0   |
+-----------------------+---------------------------------+-------------------------------+
| **Even Degree**       | Both ends point UP (↑ ... ↑)    | Both ends point DOWN (↓ ... ↓)|
| (e.g., x⁴, x⁶)        | As x → -∞, y → +∞               | As x → -∞, y → -∞             |
|                       | As x → +∞, y → +∞               | As x → +∞, y → -∞             |
+-----------------------+---------------------------------+-------------------------------+
| **Odd Degree**        | Falls Left, Rises Right (↓ ... ↑| Rises Left, Falls Right (↑ ... ↓|
| (e.g., x³, x⁵)        | As x → -∞, y → -∞               | As x → -∞, y → +∞             |
|                       | As x → +∞, y → +∞               | As x → +∞, y → -∞             |
+-----------------------+---------------------------------+-------------------------------+

Turning Points and Root Multiplicity

• Maximum Turning Points: A polynomial of degree $n$ has at most $n - 1$ turning points (local extrema).
• Multiplicity of a Zero: If $(x - c)^m$ is a factor of $P(x)$, $m$ is the multiplicity of the zero $c$:
  • Odd Multiplicity ($m = 1, 3, 5\dots$): The graph crosses the $x$-axis at $(c, 0)$. If $m \ge 3$, it flattens into an inflection point as it crosses.
  • Even Multiplicity ($m = 2, 4, 6\dots$): The graph touches and bounces off the $x$-axis at $(c, 0)$ without crossing into opposite $y$-values.

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Rational Expressions and Rational Equations

A rational expression is an algebraic fraction $\frac{P(x)}{Q(x)}$ where $Q(x) \neq 0$.

Arithmetic with Rational Expressions

1. Simplifying / Reducing: Factor both numerator and denominator completely and cancel common factors: $\frac{x^2 - 9}{x^2 + 5x + 6} = \frac{(x - 3)(x + 3)}{(x + 2)(x + 3)} = \frac{x - 3}{x + 2} \quad (x \neq -2, -3)$
2. Multiplication & Division: Multiply straight across numerators and denominators. For division, multiply by the reciprocal (invert divisor): $\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C} = \frac{A \cdot D}{B \cdot C}$
3. Addition & Subtraction: Factor denominators to determine the Least Common Denominator (LCD), multiply each term by the missing factors to achieve common denominators, and combine numerators over the shared LCD.

Solving Rational Equations & Detecting Extraneous Roots

To solve a rational equation:

1. Factor all denominators and identify all domain restrictions (values of $x$ making any denominator zero).
2. Multiply every term on both sides by the complete LCD to clear all fractions.
3. Solve the resulting linear or quadratic polynomial equation.
4. Check for Extraneous Solutions: Any calculated solution that violates an initial domain restriction (causing division by zero) must be rejected.

Worked Example: Rational Equation with Extraneous Solution

Solve for $x$: $\frac{x}{x - 3} - \frac{2}{x + 1} = \frac{8}{x^2 - 2x - 3}$

• Step 1 (Factor denominators & restrictions): $x^2 - 2x - 3 = (x - 3)(x + 1)$. Restrictions: $x \neq 3$ and $x \neq -1$.
• Step 2 (Multiply by LCD $(x - 3)(x + 1)$): $x(x + 1) - 2(x - 3) = 8$
• Step 3 (Expand and solve): $x^2 + x - 2x + 6 = 8 \implies x^2 - x + 6 = 8 \implies x^2 - x - 2 = 0$ $(x - 2)(x + 1) = 0 \implies x = 2 \quad \text{or} \quad x = -1$
• Step 4 (Verify against restrictions): $x = -1$ produces division by zero in $\frac{2}{x+1}$. Therefore, $x = -1$ is extraneous.
• Conclusion: The unique valid solution is $x = 2$.

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Radical Equations & Extraneous Root Verification

A radical equation contains a variable inside a radicand (e.g., $\sqrt{ax + b} = cx + d$).

Solving Radical Equations Step-by-Step

1. Isolate the Radical: Move all non-radical terms to the opposite side of the equation.
2. Raise Both Sides to the Index Power: If the radical is a square root, square both sides ($(\sqrt{E})^2 = E$). If expanding a binomial $(cx + d)^2$, apply FOIL: $c^2 x^2 + 2cdx + d^2$.
3. Solve the Resulting Polynomial Equation: Collect all terms to one side and solve via factoring or the quadratic formula.
4. Mandatory Check in the Original Equation: Squaring both sides is not a strictly reversible algebraic operation because $A = B \implies A^2 = B^2$, but $A^2 = B^2$ also includes $A = -B$. Therefore, candidate roots may be false (extraneous).

Worked Example: Radical Equation with Extraneous Solution

Solve for $x$: $\sqrt{3x + 7} = x + 1$.

• Step 1 (Square both sides): $(\sqrt{3x + 7})^2 = (x + 1)^2 \implies 3x + 7 = x^2 + 2x + 1$
• Step 2 (Rearrange to quadratic standard form): $x^2 - x - 6 = 0$
• Step 3 (Factor): $(x - 3)(x + 2) = 0 \implies x = 3 \quad \text{or} \quad x = -2$
• Step 4 (Check both solutions in original equation):
  • For $x = 3$: $\sqrt{3(3) + 7} = \sqrt{9 + 7} = \sqrt{16} = 4$; Right side: $3 + 1 = 4$. ($4 = 4$ is Valid).
  • For $x = -2$: $\sqrt{3(-2) + 7} = \sqrt{-6 + 7} = \sqrt{1} = 1$; Right side: $-2 + 1 = -1$. ($1 \neq -1$ is Extraneous).
• Conclusion: The only real solution is ${ 3 }$.