Polynomials - Digital SAT Math: Concepts, 18+ Practice Questions & Shortcuts

Key Concepts & Worked Archetypes

Concept 1

Concept 1: Polynomial Addition, Subtraction, and Multiplication

Combining like terms and distributing polynomials to simplify algebraic expressions or find equivalent forms.

Combine only terms with identical variable exponents: $ax^n + bx^n = (a+b)x^n$
Distribute negative signs carefully across all terms inside parentheses: $-(ax^2 + bx - c) = -ax^2 - bx + c$
Use FOIL or the distributive property for binomial multiplication: $(x+a)(x+b) = x^2 + (a+b)x + ab$

📘 Traditional Algebraic Method

Group like terms by their degree in descending order and combine their coefficients systematically.

⚡ SAT Speed Trick & Desmos Hack

Enter the original polynomial and the answer choices into Desmos as $f(x)$ and $g(x)$, then find which choice produces the exact same graph or zero difference.

💡 Worked SAT Archetype Example

Problem: Which of the following is equivalent to $(3x^2 - 5x + 2) - (2x^2 + 4x - 6)$?

📘 Step-by-Step Textbook Solution:

Step 1

Distribute the negative sign to the second polynomial: $3x^2 - 5x + 2 - 2x^2 - 4x + 6$

Step 2

Group like terms by degree: $(3x^2 - 2x^2) + (-5x - 4x) + (2 + 6)$

Step 3

Combine coefficients: $x^2 - 9x + 8$

⚡ Speed / Desmos Tactic:

Step 1

Type the expression in Desmos as $y = (3x^2 - 5x + 2) - (2x^2 + 4x - 6)$

Step 2

Type each answer choice as a separate function, e.g., $y = x^2 - 9x + 8$

Step 3

Toggle visibility to confirm the graph completely overlaps the original expression.

Concept 2

Concept 2: Factoring Quadratic and Higher-Degree Polynomials

Rewriting polynomials as a product of lower-degree factors to reveal roots and key structural properties.

Difference of squares: $a^2 - b^2 = (a-b)(a+b)$
Perfect square trinomials: $a^2 \pm 2ab + b^2 = (a \pm b)^2$
Grouping method for four-term polynomials: $ax + ay + bx + by = a(x+y) + b(x+y) = (a+b)(x+y)$

📘 Traditional Algebraic Method

Find two numbers that multiply to $ac$ and add up to $b$ for trinomials of the form $ax^2 + bx + c$.

⚡ SAT Speed Trick & Desmos Hack

Use Desmos to plot the polynomial and read the $x$-intercepts directly to instantly determine the linear factors $(x - r_1)(x - r_2)$.

💡 Worked SAT Archetype Example

Problem: Which of the following is a factor of $2x^2 + 7x - 15$?

📘 Step-by-Step Textbook Solution:

Step 1

Identify product $ac = 2(-15) = -30$ and sum $b = 7$

Step 2

Find two integers that multiply to $-30$ and add to $7$, which are $10$ and $-3$

Step 3

Rewrite the middle term: $2x^2 + 10x - 3x - 15$

Step 4

Factor by grouping: $2x(x + 5) - 3(x + 5) = (2x - 3)(x + 5)$

⚡ Speed / Desmos Tactic:

Step 1

Graph $y = 2x^2 + 7x - 15$ in Desmos

Step 2

Identify the $x$-intercepts at $x = 1.5$ (or $\frac{3}{2}$) and $x = -5$

Step 3

Convert roots to factors: $(2x - 3)$ and $(x + 5)$

Concept 3

Concept 3: Remainder Theorem and Factor Theorem

Evaluating polynomials at specific values to determine division remainders and factor existence.

Remainder Theorem: When polynomial $P(x)$ is divided by $(x - c)$, the remainder is $P(c)$
Factor Theorem: $(x - c)$ is a factor of $P(x)$ if and only if $P(c) = 0$
Polynomial Division Identity: $P(x) = Q(x) \cdot D(x) + R$

📘 Traditional Algebraic Method

Substitute $x = c$ directly into the polynomial expression or perform polynomial long/synthetic division.

⚡ SAT Speed Trick & Desmos Hack

Define $P(x)$ in Desmos using function notation, then simply evaluate $P(c)$ by typing it directly in the next line.

💡 Worked SAT Archetype Example

Problem: What is the remainder when $P(x) = 3x^3 - 4x^2 + 5x - 7$ is divided by $(x - 2)$?

📘 Step-by-Step Textbook Solution:

Step 1

Identify the root of the divisor: set $x - 2 = 0$, giving $x = 2$

Step 2

Evaluate $P(2)$ by substitution: $3(2)^3 - 4(2)^2 + 5(2) - 7$

Step 3

Simplify exponents: $3(8) - 4(4) + 10 - 7$

Step 4

Calculate final arithmetic: $24 - 16 + 10 - 7 = 11$

⚡ Speed / Desmos Tactic:

Step 1

Type $P(x) = 3x^3 - 4x^2 + 5x - 7$ in Desmos

Step 2

Type $P(2)$ in the next line

Step 3

Read output value $11$ instantly

Concept 4

Concept 4: Zeros, Roots, and Graphical Behavior

Connecting the algebraic representation of polynomials with their graphical features and multiplicities.

An odd multiplicity root causes the graph to cross the $x$-axis
An even multiplicity root causes the graph to touch and turn around at the $x$-axis
The degree of the polynomial limits the maximum number of turning points to $n-1$

📘 Traditional Algebraic Method

Set the polynomial equal to zero, factor completely, and analyze the exponent on each factor to determine turning behavior.

⚡ SAT Speed Trick & Desmos Hack

Graph the polynomial in Desmos to instantly visualize roots, turning points, and end behavior without manual factoring.

💡 Worked SAT Archetype Example

Problem: Which of the following functions has a graph that touches the $x$-axis and turns around at $x = 3$?

📘 Step-by-Step Textbook Solution:

Step 1

Recall that a graph touches and turns around at a root if the root has an even multiplicity

Step 2

Identify the factor corresponding to root $x = 3$, which is $(x - 3)$

Step 3

Apply an even exponent such as $2$ to obtain $(x - 3)^2$

⚡ Speed / Desmos Tactic:

Step 1

Test options in Desmos by graphing them one by one

Step 2

Look for the curve that kisses the $x$-axis at $x = 3$ without crossing it

Step 3

Confirm equation structure instantly

Practice Questions (18)

Question 1 Expanding and Simplifying Polynomial Expressions

Easy

Which of the following is equivalent to $(3x^2 - 5x + 2) + (2x^2 + 4x - 6)$?

📘 Step-by-Step Algebraic Solution

Step 1

Group like terms together from the expression.

Step 2

$(3x^2 + 2x^2) + (-5x + 4x) + (2 - 6)$

Step 3

Simplify the coefficients of each grouped term.

Step 4

$5x^2 - x - 4$

⚡ Desmos Shortcut / Speed Hack

Step 1

Type the given expression into Desmos as $y_1 = (3x^2 - 5x + 2) + (2x^2 + 4x - 6)$.

Step 2

Type each option as $y_2$, $y_3$, etc.

Step 3

Check which graph perfectly overlaps with $y_1$. Option A matches.

Question 2 Expanding and Simplifying Polynomial Expressions

Easy

What is the product of $(2x - 3)$ and $(x + 4)$?

📘 Step-by-Step Algebraic Solution

Step 1

Expand the product using the distributive property.

Step 2

$2x(x) + 2x(4) - 3(x) - 3(4)$

Step 3

Multiply each term out: $2x^2 + 8x - 3x - 12$

Step 4

Combine middle like terms to get $2x^2 + 5x - 12$

⚡ Desmos Shortcut / Speed Hack

Step 1

Evaluate both the expression and options at a specific value, like $x = 3$.

Step 2

Original expression value: $(2(3) - 3)(3 + 4) = (3)(7) = 21$.

Step 3

Test option B: $2(3)^2 + 5(3) - 12 = 18 + 15 - 12 = 21$. Matches.

Question 3 Expanding and Simplifying Polynomial Expressions

Medium

Which of the following is equivalent to $(x - 2)(x^2 + 2x + 4)$?

📘 Step-by-Step Algebraic Solution

Step 1

Distribute $x$ across the trinomial and $-2$ across the trinomial.

Step 2

$x(x^2 + 2x + 4) - 2(x^2 + 2x + 4)$

Step 3

$x^3 + 2x^2 + 4x - 2x^2 - 4x - 8$

Step 4

Combine like terms, noting that middle terms cancel out: $x^3 - 8$

⚡ Desmos Shortcut / Speed Hack

Step 1

Recognize this as the difference of cubes formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ where $a=x$ and $b=2$.

Step 2

Compute $x^3 - 2^3$.

Step 3

Directly obtain $x^3 - 8$.

Question 4 Expanding and Simplifying Polynomial Expressions

Medium

If $f(x) = (x^2 - 3x + 1)(2x + 3)$, what is the coefficient of $x$ in the expanded form of $f(x)$?

📘 Step-by-Step Algebraic Solution

Step 1

Identify which products yield an $x$ term when multiplying $(x^2 - 3x + 1)(2x + 3)$.

Step 2

Multiply $(-3x)(3)$ and $(1)(2x)$.

Step 3

Combine these products: $-9x + 2x$

Step 4

Simplify to find the coefficient of $x$: $-7$

⚡ Desmos Shortcut / Speed Hack

Step 1

Type $f(x) = (x^2 - 3x + 1)(2x + 3)$ into Desmos.

Step 2

Expand using a Computer Algebra System or polynomial expansion command if available, or evaluate $f'(0)$ or use finite differences.

Step 3

Alternatively, just expand manually for the linear term quickly: $-9x + 2x = -7x$.

Question 5 Expanding and Simplifying Polynomial Expressions

Hard

What is the sum of all coefficients in the expansion of the polynomial $(3x^2 - 2x + 4)^3$?

📘 Step-by-Step Algebraic Solution

Step 1

Recall that the sum of the coefficients of a polynomial $P(x)$ is found by evaluating $P(1)$.

Step 2

Substitute $x = 1$ into the expression $(3(1)^2 - 2(1) + 4)$.

Step 3

Simplify inside the parentheses: $(3 - 2 + 4) = 5$.

Step 4

Raise to the 3rd power: $5^3 = 125$

⚡ Desmos Shortcut / Speed Hack

Step 1

Define $P(x) = (3x^2 - 2x + 4)^3$ in Desmos.

Step 2

Evaluate $P(1)$ directly by typing $P(1)$ in the next line.

Step 3

Instantly read the output: 125.

Question 6 Finding Roots and Factoring Polynomials

Easy

What are all the real solutions to the equation $x^2 - 9x + 18 = 0$?

📘 Step-by-Step Algebraic Solution

Step 1

Factor the quadratic equation into two binomials: $(x - a)(x - b) = 0$.

Step 2

Identify numbers that multiply to $18$ and sum to $-9$: $-3$ and $-6$.

Step 3

Write the factored form: $(x - 3)(x - 6) = 0$.

Step 4

Set each factor to zero to find the roots: $x = 3, x = 6$

⚡ Desmos Shortcut / Speed Hack

Step 1

Graph $y = x^2 - 9x + 18$ in Desmos.

Step 2

Look at the x-intercepts on the coordinate plane.

Step 3

The x-intercepts are at $x = 3$ and $x = 6$.

Question 7 Finding Roots and Factoring Polynomials

Easy

Which of the following is a factor of the polynomial $x^2 - 16$?

📘 Step-by-Step Algebraic Solution

Step 1

Rewrite $16$ as $4^2$.

Step 2

Apply the difference of squares formula: $x^2 - 4^2 = (x - 4)(x + 4)$

Step 3

Examine the factors: $(x - 4)$ and $(x + 4)$.

Step 4

Match with the given options to find $(x + 4)$.

⚡ Desmos Shortcut / Speed Hack

Step 1

Graph $y = x^2 - 16$ and find roots $x = 4$ and $x = -4$.

Step 2

Roots $x = c$ correspond to factors $(x - c)$.

Step 3

Thus, factors are $(x - 4)$ and $(x + 4)$. Option B is correct.

Question 8 Finding Roots and Factoring Polynomials

Medium

What is the complete factorization of $2x^3 - 8x$?

📘 Step-by-Step Algebraic Solution

Step 1

Factor out the GCF, which is $2x$, from each term.

Step 2

$2x(x^2 - 4)$

Step 3

Factor the difference of squares inside the parentheses: $x^2 - 4 = (x - 2)(x + 2)$

Step 4

Combine to get the complete factorization: $2x(x - 2)(x + 2)$

⚡ Desmos Shortcut / Speed Hack

Step 1

Graph $y = 2x^3 - 8x$ in Desmos to find roots at $x = -2, 0, 2$.

Step 2

Roots give factors $(x+2)$, $x$, and $(x-2)$, with a leading coefficient of $2$.

Step 3

Match to option A: $2x(x - 2)(x + 2)$.

Question 9 Finding Roots and Factoring Polynomials

Medium

If $x = 3$ is a root of the polynomial $P(x) = x^3 - 5x^2 + kx - 9$, what is the value of $k$?

📘 Step-by-Step Algebraic Solution

Step 1

Set up the equation by substituting $x = 3$ and setting $P(3) = 0$.

Step 2

$(3)^3 - 5(3)^2 + k(3) - 9 = 0$

Step 3

Evaluate powers: $27 - 5(9) + 3k - 9 = 0 \Rightarrow 27 - 45 + 3k - 9 = 0$

Step 4

Simplify and solve for $k$: $-27 + 3k = 0 \Rightarrow 3k = 27 \Rightarrow k = 9$ wait let's re-verify: $27 - 45 = -18$, $-18 - 9 = -27$. $-27 + 3k = 0 \Rightarrow 3k = 27 \Rightarrow k=9$. Wait, let's check option B: $7$? Ah, let's recalculate: $27 - 45 = -18$. $-18 - 9 = -27$. $-27 + 3k = 0 \Rightarrow k = 9$. Let's make sure option C is 9.

⚡ Desmos Shortcut / Speed Hack

Step 1

Define $P(x) = x^3 - 5x^2 + kx - 9$ in Desmos with a slider for $k$.

Step 2

Adjust $k$ until the graph passes through the point $(3, 0)$.

Step 3

Read the value of $k$ as $9$ (Note: adjust option C to be correct, index 2).

Question 10 Finding Roots and Factoring Polynomials

Hard

The polynomial $P(x) = x^4 - 2x^3 - 7x^2 + 8x + 12$ has four real roots. What is the sum of the squares of all four roots?

Hint: Use Vieta's formulas for roots $r_1, r_2, r_3, r_4$: sum of roots is $-b/a$ and sum of products taken two at a time is $c/a$. Then use the identity $(\sum r_i)^2 = \sum r_i^2 + 2\sum_{i

📘 Step-by-Step Algebraic Solution

Step 1

Let the roots be $r_1, r_2, r_3, r_4$. From Vieta's formulas for $x^4 - 2x^3 - 7x^2 + 8x + 12 = 0$, the sum of roots $\sum r_i = -(-2)/1 = 2$.

Step 2

The sum of pairwise products $\sum_{i

Step 3

Use the identity $\sum r_i^2 = (\sum r_i)^2 - 2(\sum_{i

Step 4

Substitute values: $\sum r_i^2 = (2)^2 - 2(-7) = 4 + 14 = 18$

⚡ Desmos Shortcut / Speed Hack

Step 1

Graph $y = x^4 - 2x^3 - 7x^2 + 8x + 12$ in Desmos to find the exact roots.

Step 2

The roots are integers: $x = -2, -1, 2, 3$.

Step 3

Compute $(-2)^2 + (-1)^2 + 2^2 + 3^2 = 4 + 1 + 4 + 9 = 18$.

Question 11 Polynomial Division and Remainder Theorem

Easy

What is the remainder when the polynomial $P(x) = x^3 - 2x^2 + 5x - 4$ is divided by $(x - 2)$?

📘 Step-by-Step Algebraic Solution

Step 1

Identify $c = 2$ from the divisor $(x - 2)$.

Step 2

Substitute $x = 2$ into the polynomial $P(x)$.

Step 3

$P(2) = (2)^3 - 2(2)^2 + 5(2) - 4$

Step 4

Calculate: $8 - 2(4) + 10 - 4 = 8 - 8 + 10 - 4 = 6$

⚡ Desmos Shortcut / Speed Hack

Step 1

Define $P(x) = x^3 - 2x^2 + 5x - 4$ in Desmos.

Step 2

Type $P(2)$ in the next line.

Step 3

Instantly read the remainder: 6.

Question 12 Polynomial Division and Remainder Theorem

Easy

When $2x^2 + 3x - 5$ is divided by $(x - 1)$, what is the quotient?

📘 Step-by-Step Algebraic Solution

Step 1

Divide the first term $2x^2$ by $x$ to get $2x$.

Step 2

Multiply $2x$ by $(x - 1)$ to get $2x^2 - 2x$.

Step 3

Subtract from original: $(2x^2 + 3x) - (2x^2 - 2x) = 5x - 5$.

Step 4

Divide $5x$ by $x$ to get $+5$, giving a quotient of $2x + 5$.

⚡ Desmos Shortcut / Speed Hack

Step 1

Multiply $(x - 1)$ by each option to see which yields $2x^2 + 3x - 5$.

Step 2

Test Option A: $(x - 1)(2x + 5) = 2x^2 + 5x - 2x - 5 = 2x^2 + 3x - 5$.

Step 3

Perfect match.

Question 13 Polynomial Division and Remainder Theorem

Medium

If $\frac{x^3 - 3x^2 + 4x - 5}{x - 2} = x^2 - x + 2 + \frac{R}{x - 2}$, what is the value of the remainder $R$?

📘 Step-by-Step Algebraic Solution

Step 1

Use the Remainder Theorem by evaluating the numerator at $x = 2$.

Step 2

$R = (2)^3 - 3(2)^2 + 4(2) - 5$

Step 3

Calculate each term: $8 - 3(4) + 8 - 5$

Step 4

$8 - 12 + 8 - 5 = -1$

⚡ Desmos Shortcut / Speed Hack

Step 1

Define $f(x) = x^3 - 3x^2 + 4x - 5$ in Desmos.

Step 2

Find $f(2)$ by evaluating it directly.

Step 3

Result is $-1$.

Question 14 Polynomial Division and Remainder Theorem

Medium

Which of the following expressions is equal to $\frac{2x^3 + 3x^2 - 8x + 5}{x + 3}$?

📘 Step-by-Step Algebraic Solution

Step 1

Divide leading term $2x^3$ by $x$ to get $2x^2$.

Step 2

Multiply and subtract: $(2x^3 + 3x^2) - (2x^3 + 6x^2) = -3x^2 - 8x$.

Step 3

Divide $-3x^2$ by $x$ to get $-3x$. Multiply and subtract: $(-3x^2 - 8x) - (-3x^2 - 9x) = x + 5$.

Step 4

Divide $x$ by $x$ to get $+1$. Remainder is $(x+5) - (x+3) = 2$. Thus quotient is $2x^2 - 3x + 1$ with remainder $2$.

⚡ Desmos Shortcut / Speed Hack

Step 1

Check the remainder by evaluating the numerator at $x = -3$: $2(-3)^3 + 3(-3)^2 - 8(-3) + 5 = -54 + 27 + 24 + 5 = 2$.

Step 2

Only option A has a remainder of $\frac{2}{x + 3}$.

Step 3

Option A is correct.

Question 15 Polynomial Division and Remainder Theorem

Hard

When $P(x) = x^4 + ax^3 - 3x^2 + bx - 4$ is divided by $(x - 2)$, the remainder is $10$. When $P(x)$ is divided by $(x + 1)$, the remainder is $-8$. What is the value of $a + b$?

📘 Step-by-Step Algebraic Solution

Step 1

Apply the Remainder Theorem for $P(2) = 10$: $(2)^4 + a(2)^3 - 3(2)^2 + b(2) - 4 = 10$.

Step 2

Simplify: $16 + 8a - 12 + 2b - 4 = 10 \Rightarrow 8a + 2b = 10 \Rightarrow 4a + b = 5$.

Step 3

Apply the Remainder Theorem for $P(-1) = -8$: $(-1)^4 + a(-1)^3 - 3(-1)^2 + b(-1) - 4 = -8$.

Step 4

Simplify: $1 - a - 3 - b - 4 = -8 \Rightarrow -a - b - 6 = -8 \Rightarrow a + b = 2$. Wait let's re-verify: $-a - b = -2 \Rightarrow a + b = 2$. Let's check options: Option B is 3. Let's recalculate carefully. $(-1)^4 = 1$, $a(-1)^3 = -a$, $-3(-1)^2 = -3$, $b(-1) = -b$, constant is $-4$. Sum: $1 - a - 3 - b - 4 = -a - b - 6$. Set equal to $-8$: $-a - b - 6 = -8 \Rightarrow -a - b = -2 \Rightarrow a + b = 2$. Wait, let's fix option A or B so that $2$ or $3$ matches. Let's make option A equal to 2.

⚡ Desmos Shortcut / Speed Hack

Step 1

Set up two equations in Desmos with parameters $a$ and $b$: $16+8a-12+2b-4 = 10$ and $1-a-3-b-4 = -8$.

Step 2

Solve the linear system to find $a$ and $b$.

Step 3

Sum $a + b$ to get the result.

Question 16 End Behavior and Graph Characteristics of Polynomials

Easy

Which of the following describes the end behavior of the polynomial function $f(x) = -3x^4 + 5x^2 - x + 7$ as $x$ approaches positive and negative infinity?

📘 Step-by-Step Algebraic Solution

Step 1

Identify the leading term of the polynomial: $-3x^4$.

Step 2

Note that the degree is even ($4$) and the leading coefficient is negative ($-3$).

Step 3

An even-degree polynomial with a negative leading coefficient points downwards in both directions.

Step 4

Therefore, as $x \to \pm\infty$, $f(x) \to -\infty$.

⚡ Desmos Shortcut / Speed Hack

Step 1

Visualize or sketch a parabola opening downwards, like $y = -x^2$, which has the same end behavior type.

Step 2

Both ends point down to negative infinity.

Step 3

Select option B.

Question 17 End Behavior and Graph Characteristics of Polynomials

Easy

What is the degree of the polynomial function $P(x) = (x - 2)^2(x + 3)(2x - 1)^3$?

📘 Step-by-Step Algebraic Solution

Step 1

Identify the exponent of each linear factor in the polynomial.

Step 2

$(x - 2)$ has exponent $2$, $(x + 3)$ has exponent $1$, and $(2x - 1)$ has exponent $3$.

Step 3

Add the exponents together: $2 + 1 + 3 = 6$.

Step 4

The degree of the polynomial is $6$.

⚡ Desmos Shortcut / Speed Hack

Step 1

Mentally multiply the highest degree terms of each factor: $x^2 \cdot x^1 \cdot (2x)^3$.

Step 2

$x^2 \cdot x \cdot 8x^3 = 8x^6$.

Step 3

The highest power of $x$ is $6$.

Question 18 End Behavior and Graph Characteristics of Polynomials

Medium

How many local extrema (turning points) can a polynomial of degree $5$ have at most?

📘 Step-by-Step Algebraic Solution

⚡ Desmos Shortcut / Speed Hack

Official SAT Exam Blueprint & Weightage

Metric	Exam Pattern & Weightage
Question Frequency	1-3 questions per test module
Module 1 Appearance	Medium Frequency (Foundational tests)
Module 2 Appearance	High Frequency (Hard module score differentiator)
Pattern Analysis	Analysis of College Board question patterns and recent exam distributions.

🏛️

Official SAT PYQ Drill Bank (2023–2026)

We are compiling real verified College Board exam patterns with step-by-step Desmos shortcuts for Polynomials.

Updated for 2026 Testing Season

Revision, Traps & Desmos Syntax

Remainder Theorem

$P(c) = \text{Remainder}$

Evaluating $P(x)$ at $x=c$ yields the exact remainder when divided by $(x-c)$.

Difference of Cubes

$a^3 - b^3 = (a-b)(a^2 + ab + b^2)$

Used to factor cubic expressions into linear and irreducible quadratic terms.

Sum of Cubes

$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$

Used to factor sums of cubic terms.

🚨 Top SAT Traps & Misconceptions

⚠️ SAT Trap: Sign Distribution Errors in Subtraction

Forgetting to distribute a negative sign across all terms inside a subtracted polynomial parenthesis.

⚠️ SAT Trap: Confusing Remainder with Quotient

Reading the quotient or coefficient of $x$ instead of evaluating the constant remainder value $P(c)$.

⚡ Essential Desmos Cheatsheet

🎯 Function Evaluation Shortcut

Define $f(x) = \dots$, then evaluate $f(c)$

Instantly computes polynomial values and remainders without manual substitution arithmetic.

🎯 Root to Factor Inspection

Plot $y = P(x)$

Click $x$-intercepts $(r, 0)$ to immediately write down factors $(x - r)$.

3-Level Mock Test (30 Questions)

🟢 Level 1: Foundation

10 Qs · Sub-600 Score

🟡 Level 2: Target 700+

10 Qs · 600–740 Score

🔴 Level 3: 800-Mastery

10 Qs · 750–800 Score

Question 1 Level 1: Foundation

Which of the following expressions is equivalent to $(3x^2 + 5x - 2) + (2x^2 - x + 4)$?

Question 2 Level 1: Foundation

What is the product of the polynomials $(x + 3)$ and $(x - 4)$?

Question 3 Level 1: Foundation

If $f(x) = 2x^3 - 4x + 5$, what is the value of $f(2)$?

Question 4 Level 1: Foundation

Which of the following is a factor of the polynomial $x^2 - 9$?

Question 5 Level 1: Foundation

What is the degree of the polynomial $4x^5 - 3x^2 + 7x^6 - 2$?

Question 6 Level 1: Foundation

Subtract $(5x^2 - 2x + 3)$ from $(8x^2 + 4x - 1)$.

Question 7 Level 1: Foundation

Which of the following is equivalent to $4x(2x^2 - 3x + 5)$?

Question 8 Level 1: Foundation

What is the y-intercept of the polynomial function $f(x) = x^3 - 4x^2 + 5x - 7$?

Question 9 Level 1: Foundation

Which expression is completely factored form of $2x^2 + 8x$?

Question 10 Level 1: Foundation

If $3x - 2$ is a factor of $3x^2 + x - 10$, what is the other linear factor?

Question 1 Level 2: Target 700+

If $(x - 2)$ is a factor of the polynomial $P(x) = x^3 - 3x^2 + kx - 4$, what is the value of $k$?

Question 2 Level 2: Target 700+

Which of the following is equivalent to the expression $\frac{2x^3 - 5x^2 + 3x - 6}{x - 2}$?

Question 3 Level 2: Target 700+

What are all the real solutions to the equation $x^4 - 5x^2 + 4 = 0$?

Question 4 Level 2: Target 700+

If $ax^3 + bx^2 + cx + d = (2x - 1)(x + 3)(x - 2)$, what is the value of $b$?

Question 5 Level 2: Target 700+

The polynomial function $P(x) = 2x^3 - 9x^2 + 12x - 4$ has a known root of multiplicity 2. What is this root?

Question 6 Level 2: Target 700+

What is the remainder when $P(x) = 3x^{100} - 5x^{50} + 2$ is divided by $x + 1$?

Question 7 Level 2: Target 700+

For what value of constant $c$ does the equation $x^3 - 6x^2 + 11x - c = 0$ have roots that are consecutive integers?

Question 8 Level 2: Target 700+

Which of the following polynomials has roots at $x = 3$, $x = -2$, and $x = 1$ with a leading coefficient of $2$?

Question 9 Level 2: Target 700+

If $P(x)$ is a polynomial of degree 3 such that $P(1) = 2, P(2) = 2, P(3) = 2$, and $P(4) = 14$, what is $P(5)$?

Question 10 Level 2: Target 700+

What is the sum of all coefficients of the expanded polynomial $(2x - 3)^{4}$?

Question 1 Level 3: 800 Mastery

Let $P(x)$ be a monic polynomial of degree 4 such. If $P(1) = 1, P(2) = 4, P(3) = 9, P(4) = 16$, what is the value of $P(5)$?

Question 2 Level 3: 800 Mastery

If the polynomial $x^4 + ax^3 + bx^2 + cx + d$ is divisible by $(x-1)^3$, what is the value of $b$?

Question 3 Level 3: 800 Mastery

What is the sum of the squares of all the roots of the polynomial equation $x^3 - 5x^2 + 6x - 1 = 0$?

Question 4 Level 3: 800 Mastery

If $x^2 - x - 1 = 0$ has roots $\alpha$ and $eta$, what is the exact value of the polynomial expression $\alpha^5 + eta^5$?

Question 5 Level 3: 800 Mastery

Find the sum of all real values of $x$ that satisfy the equation $(x^2 - 5x + 5)^{x^2 - 4x + 3} = 1$.

Question 6 Level 3: 800 Mastery

If $P(x)$ is a polynomial such that $xP(x - 1) = (x - 3)P(x)$ for all real numbers $x$, what is the sum of all possible roots of $P(x) = 0$?