Rational Expressions Calculator

P/Q — Simplify

GCF(P,Q) → cancel → state excluded values x≠r

×÷ — Mul/Div

Div=KCF; mul N×N and D×D; cross-cancel; simplify

+− — Add/Sub

Find LCD; rewrite both fractions; combine N; simplify

⌈⌋ — Complex

(A/B)÷(C/D) = (A/B)×(D/C) = AD/BC; simplify

A/r — Partial

Improper: long div first. Cover-up method for Aᵢ

=0 — Rat Eq

Excl values; cross-multiply; solve; reject extraneous

≥0 — Inequality

Zeros of P and Q → sign chart → interval notation

VA — Asymptotes

Holes=GCF; VA=Q_red=0; HA=deg compare; OA=div

Simplify \(\frac{P}{Q}=\frac{P/g}{Q/g},\;g=\gcd(P,Q)\)

Multiply \(\frac{P_1}{Q_1}\cdot\frac{P_2}{Q_2}=\frac{P_1P_2}{Q_1Q_2}\)

Divide \(\frac{P_1}{Q_1}\div\frac{P_2}{Q_2}=\frac{P_1Q_2}{Q_1P_2}\)

Add \(\frac{P_1}{Q_1}+\frac{P_2}{Q_2}=\frac{P_1Q_2+P_2Q_1}{Q_1Q_2}\)

LCD \(\text{LCD}=\text{lcm}(Q_1,Q_2)\)

Complex \(\frac{A/B}{C/D}=\frac{AD}{BC}\)

Cover-Up \(A_i=\frac{P(r_i)}{\prod_{j\neq i}(r_i-r_j)}\)

Rat Eq \(P_1Q_2=P_2Q_1,\;Q_i\neq0\)

Ineq \(\text{Sign chart at zeros of }P,Q\)

Excluded \(Q(r)=0\Rightarrow x\neq r\)

Extraneous \(\text{Check in original}\;\frac{P_1}{Q_1}=\frac{P_2}{Q_2}\)

Hole \(\gcd(P,Q)=0\text{ at }x=r\)

Vert Asymp \(Q_{\text{red}}(a)=0\)

Horiz Asymp \(\deg P<\deg Q\Rightarrow y=0\)

Horiz Asymp \(\deg P=\deg Q\Rightarrow y=a_n/b_n\)

Oblique \(\deg P=\deg Q+1\Rightarrow y=\text{quotient}\)

p/qRational Expressions Calculator