Math Buddyv1.0

Polynomial equation calculator with steps

Type or scan a polynomial equation. MathBuddy factors it, finds every real root and shows each step.

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What this page covers

  • Solving polynomial equations by factoring, the rational root test and synthetic division.
  • Multiplying polynomials and collecting like terms.
  • Dividing one polynomial by another, giving the quotient and any remainder.
  • Not covered here: listing only the degree or the standard form (use Degree of polynomial calculator), and roots that are not real numbers, which you can still type and ask about.

How to enter the problem

  • Type it with ^ for powers, for example x^3 − 6x^2 + 11x − 6 = 0, or write "multiply" or "divide" before two polynomials.
  • Scan it. Fit the one problem in the frame.
  • Choose a photo. Several problems in one photo are solved as a list.
  • Then tap Solve.

Worked examples

Example 1

Problem

Multiply (2x−3)(x2+4x−1)\displaystyle (2x - 3)(x^2 + 4x - 1).

Answer

Verified

The answer is

2x3+5x2−14x+32x^3 + 5x^2 - 14x + 3

Explanation

  1. Distribute the first term
    2x(x2+4x−1)=2x3+8x2−2x2x(x^2 + 4x - 1) = 2x^3 + 8x^2 - 2x
  2. Distribute the second term
    −3(x2+4x−1)=−3x2−12x+3-3(x^2 + 4x - 1) = -3x^2 - 12x + 3
  3. Collect like terms
    2x3+(8−3)x2+(−2−12)x+32x^3 + (8 - 3)x^2 + (-2 - 12)x + 3.

Example 2

Problem

Divide x3−2x2−5x+6\displaystyle x^3 - 2x^2 - 5x + 6 by x−3\displaystyle x - 3.

Answer

Verified

The answer is

x2+x−2x^2 + x - 2

Explanation

  1. Set up synthetic division with 3
    Write the coefficients 1,−2,−5,61, -2, -5, 6.
  2. Bring down, multiply, add
    Bring down 1. 1⋅3=31 \cdot 3 = 3 and −2+3=1-2 + 3 = 1. 1⋅3=31 \cdot 3 = 3 and −5+3=−2-5 + 3 = -2. −2⋅3=−6-2 \cdot 3 = -6 and 6−6=06 - 6 = 0.
  3. Read the result
    The quotient has coefficients 1,1,−21, 1, -2 and the remainder is 0.

Example 3

Problem

Solve x4−5x2+4=0\displaystyle x^4 - 5x^2 + 4 = 0.

Answer

Verified

The answer is

x=−2,x=−1,x=1,x=2x = -2, x = -1, x = 1, x = 2

Explanation

  1. Spot the quadratic form
    Let u=x2u = x^2. The equation becomes u2−5u+4=0u^2 - 5u + 4 = 0.
  2. Factor in uu
    (u−1)(u−4)=0(u - 1)(u - 4) = 0, so u=1u = 1 or u=4u = 4.
  3. Return to xx
    x2=1x^2 = 1 gives x=±1x = \pm 1, and x2=4x^2 = 4 gives x=±2x = \pm 2.

Common mistakes

  • Dividing both sides by xx and losing the root x=0x = 0. Move everything to one side and factor instead.
  • Leaving out a placeholder zero in synthetic division when a power is missing, such as the x2x^2 term of x3−4x+1x^3 - 4x + 1.
  • Keeping only the positive root after a substitution. x2=4x^2 = 4 has two roots, 2 and −2-2.
  • Dropping a sign when distributing: −3(x2+4x−1)-3(x^2 + 4x - 1) ends in +3+3, so the product ends in +3+3, not −3-3.

Checks, assumptions and limits

  • Substitute each root into the equation. For the example, x=2x = 2 gives 8−24+22−6=08 - 24 + 22 - 6 = 0.
  • A polynomial of degree nn has at most nn real roots.
  • Check a division by multiplying back: (x−3)(x2+x−2)=x3−2x2−5x+6(x - 3)(x^2 + x - 2) = x^3 - 2x^2 - 5x + 6.
  • MathBuddy can make mistakes. Double check important steps.

Related tasks

Frequently asked questions

Yes, when the roots can be reached by factoring, the rational root test or a substitution such as u = x^2. The steps say which method is used.

Yes. Ask to divide by a linear factor such as x − 3, and the steps lay out the coefficients and each multiply-and-add.

It is given exactly, as a fraction or a root such as √2. Ask a follow-up for a decimal approximation.

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