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| Example polynomial | Factored form | Main method |
|---|---|---|
| x2 + 5x + 6 | (x + 2)(x + 3) | Quadratic factoring |
| 2x2 + x - 1 | (2x - 1)(x + 1) | Rational roots |
| 4x4 - 23x2 + 15 | (x2 - 5)(4x2 - 3) | Substitution with y = x2 |
| x3 - 6x2 + 11x - 6 | (x - 1)(x - 2)(x - 3) | Synthetic division |
General polynomial: P(x) = anxn + an-1xn-1 + ... + a1x + a0.
Common factor extraction: P(x) = g xm Q(x), where g is the greatest shared coefficient factor and xm is the lowest shared variable power.
Grouping idea: A + B + C + D = G1H + G2H = (G1 + G2)H when a common grouped factor H appears.
Rational Root Theorem: if p/q is a rational root, then p divides the constant term and q divides the leading coefficient.
Quadratic test: x = [-b ± √(b2 - 4ac)] / 2a. A perfect-square discriminant often means rational linear factors exist.
Exponent substitution: if all exponents share k, let y = xk. Factor the smaller polynomial in y, then substitute back.
Question: Factor the polynomial 4x4 – 20x2 – 3x2 + 15 by grouping. What is the resulting expression?
Answer: First combine the middle like terms: 4x4 - 20x2 - 3x2 + 15 = 4x4 - 23x2 + 15.
Now group it as (4x4 - 20x2) + (-3x2 + 15) = 4x2(x2 - 5) - 3(x2 - 5).
So the resulting factored expression is (x2 - 5)(4x2 - 3).
It handles coefficient-based polynomials from degree 1 through degree 4. It looks for numeric common factors, shared powers, rational roots, substitution patterns, and quadratic leftovers.
Yes. Decimal inputs are scaled internally so the factoring rules can still check common factors and rational-root candidates more reliably, then the displayed numeric factor is converted back.
Some polynomials are prime over rational coefficients, or they need advanced symbolic methods beyond the implemented rules. In that case, the calculator still shows the cleaned expression and graph.
Yes, especially when grouped terms reveal a repeated factor after simplification. The worked example section shows how grouped quartic terms can collapse into two clear polynomial factors.
The graph helps you see where the polynomial crosses or touches the x-axis. Those intercepts often match real roots and confirm whether the final factorization behaves as expected.
Enter zero for any missing coefficient. For example, x4 - 5x2 + 4 uses zero for the x3 and x terms.
Combine the middle terms first to get 4x4 - 23x2 + 15. Then group as 4x2(x2 - 5) - 3(x2 - 5), so the result is (x2 - 5)(4x2 - 3).
This version is coefficient-driven for accuracy and cleaner factoring steps. If you have an expression, rewrite it into coefficients first, including zeros for any missing powers.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.