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Polynomials: how to simplify, expand and solve them

What a polynomial is, how to simplify and expand polynomials, and how to solve polynomial equations by factoring — with checked examples.

2 min readUpdated 2 November 2026By Sumzo team (AI-assisted)Reviewed by Sumzo AI (owner-delegated review)

A polynomial is a sum of terms made of numbers and whole-number powers of a variable, such as 2x3−5x+72x^3 - 5x + 7. To solve a polynomial equation means to find the values of xx that make it equal zero. Here is how to solve polynomials in practice: simplify, write it in standard form, factor, and set each factor to zero.

Polynomial vocabulary

  • Terms: the parts added together — 2x32x^3, −5x-5x and 77.
  • Degree: the highest power of xx — here 3, so it is a cubic.
  • Leading coefficient: the number in front of the highest power — here 2.
  • Standard form: like terms combined, powers from highest to lowest.

How to simplify polynomials

Simplifying polynomials means combining like terms — terms with exactly the same power of xx. When subtracting a bracket, change the sign of every term inside it first.

Worked example

Easy: simplify by combining like terms

Simplify (3x2+5x−2)−(x2−4x+6)(3x^2 + 5x - 2) - (x^2 - 4x + 6).

Show the working
  1. Remove the brackets; the minus changes every sign in the second one.
    3x2+5x−2−x2+4x−63x^2 + 5x - 2 - x^2 + 4x - 6
  2. Combine like terms.
    (3x2−x2)+(5x+4x)+(−2−6)=2x2+9x−8(3x^2 - x^2) + (5x + 4x) + (-2 - 6) = 2x^2 + 9x - 8

Answer: 2x2+9x−82x^2 + 9x - 8

Checked by Sumzo’s math engine

Polynomial expansion: how to expand a polynomial

To expand a product of polynomials, multiply every term in the first bracket by every term in the second, then simplify.

Worked example

Medium: expand a product

Expand (2x+3)(x2−x+4)(2x + 3)(x^2 - x + 4).

Show the working
  1. Multiply each term of x2−x+4x^2 - x + 4 by 2x2x, then by 3.
    2x3−2x2+8x+3x2−3x+122x^3 - 2x^2 + 8x + 3x^2 - 3x + 12
  2. Combine like terms.
    2x3+x2+5x+122x^3 + x^2 + 5x + 12

Answer: 2x3+x2+5x+122x^3 + x^2 + 5x + 12

Checked by Sumzo’s math engine

How to solve polynomial equations

  1. Move everything to one side so the equation reads P(x)=0P(x) = 0.
  2. Take out any common factor.
  3. Factor: try grouping, or test whole-number candidates — any whole-number root divides the constant term.
  4. Set each factor equal to zero; use the quadratic formula for a quadratic factor that won’t factor.
  5. Check each root in the original equation.

Worked example

Hard: a cubic by testing a root

Solve x3−4x2−7x+10=0x^3 - 4x^2 - 7x + 10 = 0.

Show the working
  1. Try x=1x = 1: 1−4−7+10=01 - 4 - 7 + 10 = 0, so (x−1)(x - 1) is a factor.
  2. Divide by (x−1)(x - 1) to get a quadratic.
    x3−4x2−7x+10=(x−1)(x2−3x−10)x^3 - 4x^2 - 7x + 10 = (x - 1)(x^2 - 3x - 10)
  3. Factor the quadratic.
    (x−1)(x−5)(x+2)=0(x - 1)(x - 5)(x + 2) = 0
  4. Set each factor to zero.
    x=1,  x=5,  x=−2x = 1, \; x = 5, \; x = -2

Answer: x=−2x = -2, x=1x = 1 or x=5x = 5

Checked by Sumzo’s math engine

Worked example

Hard: solve by grouping

Solve 2x3+3x2−8x−12=02x^3 + 3x^2 - 8x - 12 = 0.

Show the working
  1. Group in pairs and take out a common factor from each.
    x2(2x+3)−4(2x+3)x^2(2x + 3) - 4(2x + 3)
  2. Factor out (2x+3)(2x + 3), then the difference of two squares.
    (2x+3)(x2−4)=(2x+3)(x−2)(x+2)(2x + 3)(x^2 - 4) = (2x + 3)(x - 2)(x + 2)
  3. Set each factor to zero.
    x=−32,  x=2,  x=−2x = -\tfrac{3}{2}, \; x = 2, \; x = -2

Answer: x=−2x = -2, x=−32x = -\frac{3}{2} or x=2x = 2

Checked by Sumzo’s math engine

Common mistakes

  • Combining unlike terms: 3x2+2x3x^2 + 2x does not simplify.
  • Forgetting to change every sign when subtracting a bracket.
  • Missing terms when expanding: two terms times three terms gives six products.
  • Dividing both sides by xx, which loses the solution x=0x = 0 — factor out xx instead.

Frequently asked questions

How do you solve polynomials with a degree higher than 2?

Factor them. Look for a common factor or grouping first; otherwise test small whole numbers (divisors of the constant term) until one gives zero, divide it out, and repeat until a quadratic is left.

What is the difference between simplifying and solving a polynomial?

Simplifying rewrites the expression in a shorter form; its value doesn’t change. Solving means finding the values of xx that make P(x)=0P(x) = 0.

Can a polynomial have no real solutions?

Yes — x2+1=0x^2 + 1 = 0 has none, because x2x^2 is never negative. Its solutions are the complex numbers ii and −i-i.

Calculators

Practice

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