Key takeaway: This precalculus polynomial rational lesson shows how to list possible zeros, find every complex root, and read the graph of a rational function. The Rational Root Theorem gives candidates \(\frac{p}{q}\), and the Fundamental Theorem of Algebra promises exactly \(n\) zeros for degree \(n\). The notes follow standards A-APR.B.2 and F-IF.C.7d. In addition, they end with sign charts that solve polynomial and rational inequalities.
The Rational Root Theorem lists every possible rational zero p/q of an integer-coefficient polynomial, and the Fundamental Theorem of Algebra says a degree n polynomial has exactly n complex zeros, with nonreal zeros of real polynomials arriving as conjugate pairs. Sign charts then solve polynomial and rational inequalities.
Precalculus Polynomial Rational Lesson: key facts
- Standards A-APR.B.2, A-APR.B.3, F-IF.C.7c, F-IF.C.7d and N-CN.C.9 covered in depth at the Precalculus level.
- Rational zeros have the form \(\frac{p}{q}\) with \(p\) dividing the constant term and \(q\) dividing the leading coefficient.
- A degree \(n\) polynomial has exactly \(n\) complex zeros counting multiplicity; real polynomials have conjugate pairs \(a\pm bi\).
- End behavior follows the leading term; a slant asymptote exists when the numerator’s degree is one more than the denominator’s.
- Solve inequalities by finding critical numbers, testing intervals on a sign chart, and checking endpoints.
What does the Rational Root Theorem say?
Suppose a polynomial has integer coefficients. Then every rational zero, written in lowest terms as \(\frac{p}{q}\), has \(p\) dividing the constant term and \(q\) dividing the leading coefficient. The theorem gives a finite list of candidates to test.
For \(P(x)=2x^{3}-3x^{2}-8x+12\), \(p\) is one of \(\pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 12\) and \(q\) is \(1\) or \(2\). Testing \(x=2\) gives \(16-12-16+12=0\), so \(x=2\) is a zero. Dividing leaves \(2x^{2}+x-6=(2x-3)(x+2)\), so the zeros are \(2\), \(\frac{3}{2}\) and \(-2\).
What does the Fundamental Theorem of Algebra guarantee?
A polynomial of degree \(n \ge 1\) has exactly \(n\) complex zeros when each zero is counted as many times as its multiplicity. If the coefficients are real, the nonreal zeros come in conjugate pairs \(a+bi\) and \(a-bi\). For \(x^{2}-4x+13\) the discriminant is \(-36\), so the zeros are \(2+3i\) and \(2-3i\).
How do you describe end behavior?
For large \(|x|\) a polynomial behaves like its leading term \(a_{n}x^{n}\). Use the degree and the sign of the leading coefficient.
| Degree | Leading coefficient | As x → −∞ | As x → ∞ |
|---|---|---|---|
| even | positive | rises | rises |
| even | negative | falls | falls |
| odd | positive | falls | rises |
| odd | negative | rises | falls |
A zero of even multiplicity, such as \(x=1\) below, touches the axis and turns; an odd multiplicity crosses.
How do you find a slant asymptote?
When the numerator’s degree is exactly one more than the denominator’s, divide. The quotient is the slant asymptote and the remainder term fades as \(|x|\) grows. For \(f(x)=\frac{x^{2}-x-2}{x-1}\), division gives \(f(x)=x-\frac{2}{x-1}\), so the slant asymptote is \(y=x\) and the vertical asymptote is \(x=1\).
How do you solve polynomial and rational inequalities?
- Move every term to one side so the other side is \(0\).
- Factor the expression.
- Mark the critical numbers: zeros of the numerator and values that make the denominator zero.
- Test one value in each interval to find its sign.
- Write the intervals that match; include an endpoint only for \(\le\) or \(\ge\) and only if the expression is defined there.
Example: \((x+2)(x-1)(x-3) \ge 0\). At \(x=0\) the product is \(6\), which is positive, and the signs alternate across single zeros. The solution is \([-2,1]\) ∪ \([3,\infty)\).
Polynomial and Rational Functions in Depth: frequently asked questions
What are the possible rational zeros of a polynomial?
Take every divisor \(p\) of the constant term and every divisor \(q\) of the leading coefficient, then form \(\pm\frac{p}{q}\). For \(3x^{3}+x-8\) the candidates have \(p\) in \(1,2,4,8\) and \(q\) in \(1,3\).
Why do complex zeros come in pairs?
For a polynomial with real coefficients, the quadratic formula produces \(\pm\sqrt{b^{2}-4ac}\) when the discriminant is negative, so \(a+bi\) and \(a-bi\) always appear together.
How can you tell if a graph has a slant asymptote?
Its equation is a rational function whose numerator has degree exactly one more than its denominator. Long division gives the line, and the graph approaches that line as \(x\) grows large in either direction.
When do you include endpoints in an inequality answer?
Include a critical number only when the inequality has an equals sign and the expression is defined there. Zeros of a denominator are never included.
How does multiplicity change a graph?
An odd multiplicity makes the graph cross the x-axis, while an even multiplicity makes it touch and turn. In \((x-2)^{2}(x+1)\), the zero \(2\) touches and \(-1\) crosses.
How do you list the possible rational zeros?
Take \(P(x)=2x^3+x^2-13x+6\). The constant term 6 has factors \(\pm1,\pm2,\pm3,\pm6\), and the leading coefficient 2 has factors \(\pm1,\pm2\). Dividing gives 12 different candidates, including \(\pm\frac{1}{2}\) and \(\pm\frac{3}{2}\). Then test them with synthetic division until the remainder is zero.
Why do complex zeros come in pairs?
A polynomial with real coefficients cannot have a lone nonreal zero. If \(2-i\) is a zero, then \(2+i\) must be one too. For example, the pair gives the factor \(x^2-4x+5\). Therefore a degree 5 polynomial always keeps at least one real zero.
What is the precalculus polynomial rational lesson rule for asymptotes?
Compare degrees. When the numerator is one degree higher, divide to get a slant line. For \(\frac{x^2+x-2}{x-2}\), division gives \(x+3\) plus a remainder term, so the slant asymptote is \(y=x+3\). When degrees match, instead use the ratio of leading coefficients.
How do you solve a rational inequality?
Move everything to one side, factor, and mark the zeros and the undefined points. Next, test one value in each interval. Always keep a denominator zero open, even when the sign is \(\ge\). Otherwise your answer includes a point where the function does not exist.
Where should your polynomial practice go next?
Build three polynomials from given zeros, one with a conjugate pair, and expand each into standard form. Then sketch a rational graph with one slant asymptote. Afterward, work through the worksheet to lock in factoring, and use the quiz to see which theorem still feels shaky.
- Polynomial and Rational Functions in Depth worksheet (24 problems)
- Polynomial and Rational Functions in Depth quiz (10 questions)
- Polynomial and Rational Functions in Depth test (45 minutes)
- Previous lesson: Functions: Domain, Range, Composition and Inverses
- Next lesson: Exponential and Logarithmic Functions and Models
See every chapter on the Precalculus math page.
