Digital SAT · Math · Unit 2 · Advanced Math

Lesson 2.1 · Polynomials

On the test: 1–3 questions a test, often the hard last questions of a module

Big idea. A polynomial is a sum of terms like 3x² or −5x. Written as a PRODUCT of factors, it tells you everything: each factor (x − r) is a zero, an x-intercept, and a place where the graph changes sign. Most polynomial questions are just ‘switch between the sum form and the factor form’.

🎛 Move it · factor or not?

August US 2023: $x^{2} - 5 x - 24$ . Pick each option; only the true factorization draws the SAME curve.

Target: $x^{2} - 5 x - 24$

drag x: 1

target -28A -21

✗ Two curves — they split apart. Find an x where the values differ and the option is dead.

🧑‍🏫 Rule 1 · Factor a trinomial x² + bx + cstep 1 of 5

1
the question
$x^{2} - 5 x - 24$
We want $(x + p) (x + q)$ . Multiplying it out gives $x^{2} + (p + q) x + pq$ .

🧑‍🏫 Rule 2 · “Which expression is a factor?” — take out the common factor firststep 1 of 3

1
📋 The question · SAT · August 2024 · Math 800 Book Q10
Which expression is a factor of $2 x^{2} + 30 x + 8$ ?
1. A)2
2. B) $4 x$
3. C) $30 x$
4. D) $2 x^{2}$
the question
$2 x^{2} + 30 x + 8$
August 2024: which is a factor? Options: 2, $4 x$ , $30 x$ , $2 x^{2}$ .

🧑‍🏫 Rule 3 · A factor with a letter in itstep 1 of 5

1
📋 The question · SAT · May 2024 · Math 800 Book Q7
Which expression has a factor of $x + 2 b$ , where $b$ is a positive integer constant?
1. A) $3 x^{2} + 9 x + 18 b$
2. B) $3 x^{2} + 24 x + 18 b$
3. C) $3 x^{2} + 30 x + 18 b$
4. D) $3 x^{2} + 39 x + 18 b$
the question
$factor x + 2 b of 3 x^{2} +__x + 18 b$
May 2024: $b$ is a positive integer. Which middle number works?

Desmos move · check a factorization

Graph the original and an option. One curve → equivalent. Or graph the original alone: its

x

-intercepts are the zeros, so

x = - 3

and

x = 8

give

(x + 3) (x - 8)

🎛 Move it · build it from its zeros

Place up to three zeros. Watch: each one is a factor in the formula, a crossing on the graph, and a place where the sign band switches.

−

f (x) = (x + 3) (x - 1) (x - 4)

zero 1-3zero 21zero 34

Each zero r gives a factor (x − r) and an x-intercept (r, 0). Sliding a zero slides that crossing — nothing else.

f(0) = a × (−r₁)(−r₂)… = 12. The sign flips at every zero.

🧑‍🏫 Rule 4 · Zero ⇔ factor ⇔ x-interceptstep 1 of 3

1
the idea
$f (r) = 0 ⟺ (x - r) is a factor ⟺ (r, 0) on the graph$
Three ways of saying the same thing. A product is 0 exactly when one factor is 0.

🧑‍🏫 Rule 5 · Use a known zero to find a missing constantstep 1 of 4

1
📋 The question · SAT · June 2024 · Math 800 Book Q3
$f (x) = (x - 5) (x - 8) (x + k)$ , where $k$ is a constant. The graph passes through $(- 2, 0)$ . What is $f (0)$ ?
1. A) $- 140$
2. B) $- 2$
3. C) $11$
4. D) $80$
the question
$f (x) = (x - 5) (x - 8) (x + k), (- 2, 0) on the graph$
June 2024: find $f (0)$ .

🧑‍🏫 Rule 6 · Sum and product of the solutionsstep 1 of 5

1
📋 The question · SAT · May 2024 · Math 800 Book Q4
$7 x^{2} (4 x - 13) (4 x - u) = 0$
In the given equation, $u$ is a positive constant. The sum of the solutions is 9. What is $u$ ?
(type the answer)
the question
$7 x^{2} (4 x - 13) (4 x - u) = 0$
May 2024: the sum of the solutions is 9. Find $u$ .

Real question 1SAT · June 2024 · Math 800 Book Q3

The function $f$ is defined by $f (x) = (x - 5) (x - 8) (x + k)$ , where $k$ is a constant. In the $x y$ -plane, the graph of $y = f (x)$ passes through the point $(- 2, 0)$ . What is the value of $f (0)$ ?

🧑‍🏫 Rule 7 · The sign changes at each zerostep 1 of 4

1
📋 The question · SAT · August 2024 · Math 800 Book Q12
$f (x) = 3 (x - a) (x - b) (x - c)$
$a$ , $b$ , $c$ are distinct constants. When $a < x < b$ , $f (x)$ is positive. The graph contains $(r, s)$ . If $s = 8$ , which could be true? I. $r < a$ II. $b < r < c$ III. $r > c$
1. A)I only
2. B)III only
3. C)I and III only
4. D)II and III only
the question
$f (x) = 3 (x - a) (x - b) (x - c)$
August 2024: $f$ is positive when $a < x < b$ . If $(r, 8)$ is on the graph, where can $r$ be?

🧑‍🏫 Rule 8 · Shiftsstep 1 of 4

1
📋 The question · SAT · October US 2024 · Math 800 Book Q14
$f (x) = (x - 8) (x - 5) (x - 3)$ . The graph of $y = g (x)$ is the graph of $y = f (x)$ translated up 3 units. What is $g (0)$ ?
(type the answer)
up
$g (x) = f (x) + 3 \Rightarrow g (0) = f (0) + 3$
October US 2024: $f (0) = (- 8) (- 5) (- 3) = - 120$ , so $g (0) = - 117$ .

Desmos move · see the sign

Graph

f

. Wherever the curve is above the

x

-axis

f

is positive. A point

(r, 8)

must sit where the curve is above the axis.

🧑‍🏫 Rule 9 · Mirror points give equal outputsstep 1 of 4

1
📋 The question · SAT · December 2024 · Math 800 Book Q21
$p (x) = a ((x + 5)^{2} - b) ((x + 5)^{2} - c)$ . The graph passes through $(- 6, 30)$ and $(0, 342)$ . What is $p (- 10) + p (- 4)$ ?
(type the answer)
the question
$p (x) = a ((x + 5)^{2} - b) ((x + 5)^{2} - c)$
December 2024: through $(- 6, 30)$ and $(0, 342)$ . Find $p (- 10) + p (- 4)$ .

Why it works · why this is allowed

The input only ever appears inside

(x + 5)^{2}

. Two inputs with the same

(x + 5)^{2}

are indistinguishable to the function — so their outputs must match.

Real question 2SAT · August 2024 · Math 800 Book Q11type the answer

The function $p$ is defined by $p (x) = ((x + 6)^{2} - b) ((x + 6)^{2} - c)$ , where $b$ and $c$ are constants. The graph of $y = p (x)$ passes through $(- 7, 24)$ and $(0, 899)$ . What is the value of $p (- 12) + p (- 5)$ ?

🧑‍🏫 Rule · Remainder theoremstep 1 of 3

1
the rule
$p (x) \div (x - a) \Rightarrow remainder = p (a)$
Write $p (x) = (x - a) q (x) + R$ ; put $x = a$ and the first part vanishes.

🧑‍🏫 Rule · Multiplicity: cross or bouncestep 1 of 3

1
odd power
$(x - 2)^{1}, (x - 2)^{3}$
The graph CROSSES the axis at 2.

🧑‍🏫 Rule · End behaviour comes from the leading termstep 1 of 2

1
even degree
$a x^{2}, a x^{4} : both ends same way$
Up on both sides if $a > 0$ , down on both if $a < 0$ .

Take with you

▸Trinomial $x^{2} + b x + c$ : two numbers that multiply to $c$ and add to $b$ .
▸Take out a common factor first; a factor must divide EVERY term.
▸Zero ⇔ factor $(x - r)$ ⇔ $x$ -intercept $(r, 0)$ . A point $(a, b)$ means $f (a) = b$ .
▸Sum/product of solutions: set each factor to 0, list the solutions, then add or multiply.
▸The sign flips at each zero; far right, a positive leading number gives positive values.
▸ $f (x) + k$ moves up; $f (x - h)$ moves RIGHT by $h$ .
▸Built from $(x + h)^{2}$ ? Inputs mirrored about $- h$ give equal outputs.

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