Simplifying Polynomials — Practice Questions

Below are 10 practice questions on simplifying polynomials. In each question, expand the polynomial expression, combine like terms, and write the final answer in standard form (terms ordered from highest degree to lowest degree). Complete step-by-step solutions are provided directly beneath each question — ideal for algebra and precalculus students.

Question 1 — Distribute and Combine (Linear)

Simplify: \( 2(x + 3) - (-3x + 7) \)

View Solution

Distribute \(2\) and the negative sign:

\[ 2x + 6 + 3x - 7 \]

Group and combine like terms:

\[ (2x + 3x) + (6 - 7) = 5x - 1 \]

Answer: \( 5x - 1 \)

Question 2 — Distribute into Three Groups

Simplify: \( -4(8x - 10) - 3(7x - 1) + 3(4x) + 12 \)

View Solution

Distribute each coefficient:

\[ -32x + 40 - 21x + 3 + 12x + 12 \]

Group and combine like terms:

\[ (-32x - 21x + 12x) + (40 + 3 + 12) = -41x + 55 \]

Answer: \( -41x + 55 \)

Question 3 — Distribute a Negative Across a Quadratic

Simplify: \( 2(x^2 - 3x - 2) - (-3x^2 + 7x - 1) \)

View Solution
\[ (2x^2 - 6x - 4) + (3x^2 - 7x + 1) \] \[ (2x^2 + 3x^2) + (-6x - 7x) + (-4 + 1) = 5x^2 - 13x - 3 \]

Answer: \( 5x^2 - 13x - 3 \)

Question 4 — Cubic with Two Distributions

Simplify: \( -3(x^3 - x^2 - 2x - 5) - (4x^3 - 7x - 1) \)

View Solution
\[ (-3x^3 + 3x^2 + 6x + 15) - 4x^3 + 7x + 1 \] \[ (-3x^3 - 4x^3) + 3x^2 + (6x + 7x) + (15 + 1) \] \[ -7x^3 + 3x^2 + 13x + 16 \]

Answer: \( -7x^3 + 3x^2 + 13x + 16 \)

Question 5 — FOIL Twice and Subtract

Simplify: \( (-2x - 5)(2x - 8) - (2x - 5)(3x + 7) \)

View Solution

Part 1: \((-2x-5)(2x-8) = -4x^2 + 6x + 40\)

Part 2: \((2x-5)(3x+7) = 6x^2 - x - 35\)

\[ (-4x^2 + 6x + 40) - (6x^2 - x - 35) \] \[ -4x^2 + 6x + 40 - 6x^2 + x + 35 \] \[ -10x^2 + 7x + 75 \]

Answer: \( -10x^2 + 7x + 75 \)

Question 6 — Binomial Times Binomial Minus Monomial Product

Simplify: \( (x^2 + 4)(x - 4) - (-x - 1)(6x) \)

View Solution

Part 1: \((x^2+4)(x-4) = x^3 - 4x^2 + 4x - 16\)

Part 2: \((-x-1)(6x) = -6x^2 - 6x\)

\[ (x^3 - 4x^2 + 4x - 16) - (-6x^2 - 6x) \] \[ x^3 - 4x^2 + 4x - 16 + 6x^2 + 6x \] \[ x^3 + 2x^2 + 10x - 16 \]

Answer: \( x^3 + 2x^2 + 10x - 16 \)

Question 7 — Quartic with Two Products

Simplify: \( (-x^3 + 4)(x + 2) - (2x - 1)(5x) \)

View Solution

Part 1: \((-x^3+4)(x+2) = -x^4 - 2x^3 + 4x + 8\)

Part 2: \((2x-1)(5x) = 10x^2 - 5x\)

\[ (-x^4 - 2x^3 + 4x + 8) - (10x^2 - 5x) \] \[ -x^4 - 2x^3 - 10x^2 + (4x + 5x) + 8 \] \[ -x^4 - 2x^3 - 10x^2 + 9x + 8 \]

Answer: \( -x^4 - 2x^3 - 10x^2 + 9x + 8 \)

Question 8 — Binomial Product Minus Monomial Times Binomial

Simplify: \( (x^2 + 5)(x + 5) - 3x^2(3x + 1) \)

View Solution

Part 1: \((x^2+5)(x+5) = x^3 + 5x^2 + 5x + 25\)

Part 2: \(3x^2(3x+1) = 9x^3 + 3x^2\)

\[ (x^3 + 5x^2 + 5x + 25) - (9x^3 + 3x^2) \] \[ -8x^3 + 2x^2 + 5x + 25 \]

Answer: \( -8x^3 + 2x^2 + 5x + 25 \)

Question 9 — Two Products Combined with Addition

Simplify: \( (x^3 + 1)(-2x + 2) + 2x^2(4x^2 + 1) \)

View Solution

Part 1: \((x^3+1)(-2x+2) = -2x^4 + 2x^3 - 2x + 2\)

Part 2: \(2x^2(4x^2+1) = 8x^4 + 2x^2\)

\[ (-2x^4 + 2x^3 - 2x + 2) + (8x^4 + 2x^2) \] \[ 6x^4 + 2x^3 + 2x^2 - 2x + 2 \]

Answer: \( 6x^4 + 2x^3 + 2x^2 - 2x + 2 \)

Question 10 — Degree-5 Polynomial Simplification

Simplify: \( (x^4 - 3)(3x - 4) + 2x^3(2x^2 - 2) \)

View Solution

Part 1: \((x^4-3)(3x-4) = 3x^5 - 4x^4 - 9x + 12\)

Part 2: \(2x^3(2x^2-2) = 4x^5 - 4x^3\)

\[ (3x^5 - 4x^4 - 9x + 12) + (4x^5 - 4x^3) \] \[ (3x^5 + 4x^5) - 4x^4 - 4x^3 - 9x + 12 \] \[ 7x^5 - 4x^4 - 4x^3 - 9x + 12 \]

Answer: \( 7x^5 - 4x^4 - 4x^3 - 9x + 12 \)

Frequently Asked Questions

What does it mean to simplify a polynomial?

Simplifying means expanding all products, removing parentheses (carefully distributing negative signs), and combining like terms so the polynomial is written in standard form.

What is standard form for a polynomial?

Standard form lists terms in descending order by degree: \(a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0\).

How does the FOIL method work?

FOIL stands for First, Outer, Inner, Last. It's a shortcut for multiplying two binomials by distributing each term of the first binomial over the second.

What is the most common mistake when simplifying?

Forgetting to distribute a negative sign across every term inside the parentheses. Always apply the minus to each term, not just the first one.

What are like terms?

Like terms have the same variable raised to the same power. For example, \(3x^2\) and \(-5x^2\) are like terms, but \(3x^2\) and \(3x^3\) are not.