Polynomials & Factoring — 700-800 - Complete Interactive Lesson

Part 1: The 700-800 Patterns

Polynomials & Factoring: The 700-800 Patterns

Part 1 of 3 — The Archetypes Hard-Tier Items Are Built From

Archetype 1: The Factor Theorem With a Parameter

" $x + 3$ is a factor of $p(x) = 2x^{3} + ax^{2} - 13x + 6$ " means $p(-3) = 0$ — one substitution finds $a$ . Then divide out the known factor and work with the leftover quadratic. The question asks about the OTHER zeros (their sum, a product, a difference), and the parameter value itself is planted as a wrong answer.

Worked example. $p(-3) = -54 + 9a + 39 + 6 = 9a - 9 = 0$ gives . Dividing by leaves , so the other zeros sum to by Vieta — no need to find them.

Archetype 2: Vieta Sees Through the Parameter

For $ax^{3} + bx^{2} + cx + d$ : sum of zeros $= -\frac{b}{a}$ , product . The killer setup: " is a factor of ; what is the sum of the zeros?" You CAN find — but the sum is . Spotting that the parameter is a decoy converts a 3-minute item into a 15-second one.

Archetype 3: Special Patterns

Difference of squares, twice: $16x^{4} - 81 = (4x^{2} - 9)(4x^{2} + 9)$ , and only the first factor has real zeros.

Archetype 4: Factored Cubics in Context

$C(x) = x(2x - 3)(x - 4)$ models something; zeros are read off the factors ( $0$ , $\frac{3}{2}$ , ) — but the stem measures in HUNDREDS of units, asks for a difference rather than a value, or both. The algebra is the easy half; the last sentence is the hard half.

Part 2: Traps & Speed

Polynomials & Factoring: Traps & Speed

Part 2 of 3 — Distractor Autopsy and Faster Routes

The Four Standard Distractors

The parameter itself. The value of $k$ or $a$ found in step one is always a choice. It is never the answer — it is the toll you paid to start.
The sign-read of factors. $(2x + 5)$ has zero $-\frac{5}{2}$ , not . Zeros listed with flipped signs form a full distractor set.

Part 3: Timed Drill

Polynomials & Factoring: Timed Drill

Part 3 of 3 — Four Questions at Full Difficulty

Budget about 75 seconds per question. Opening moves: named factor $\rightarrow$ substitute its zero; remainder $\rightarrow$ evaluate, never divide; sum/product of zeros $\rightarrow$ Vieta; quartic or cubic constant patterns $\rightarrow$ look for difference of squares or cubes before anything else. Save the last 10 seconds to reread which quantity — and which UNIT — the stem wants.

$P(1)$ is the sum of the coefficients. "Remainder when divided by $x - 1$ ," "sum of coefficients of the quotient" — both collapse to evaluations at $1$ . For a clean division $P(x) = D(x)Q(x)$ : $Q(1) = \frac{P(1)}{D(1)}$ .
Remainder theorem over long division. Remainder of $P(x) \div (x - c)$ is $P(c)$ . Long division is for finding QUOTIENTS only — and even then, synthetic division is faster.
Vieta before dividing. Asked for a sum or product of remaining zeros? Total (from Vieta) minus the known zero. Division is only needed when you must know the zeros individually.
Desmos exploit: graph the polynomial and read the zeros directly; for a parameter, add a slider and tune until the graph passes through the required zero. Exact fractions still deserve an algebra check.

Polynomials & Factoring — 700-800