Unit 2 · Topic 1.11
Equivalent Representations of Polynomial and Rational Expressions
Choose among factored, standard, and quotient-remainder forms to reveal the information needed, then use long division and the binomial theorem to move accurately between forms.
1. Equivalent Expressions Versus Equal Functions
Equivalent expressions produce the same value wherever both expressions are defined. To define the same function, they must also have the same domain.
\[\frac{(x-3)(x+2)}{x-3}=x+2\quad\text{for }x\ne3.\]
The formulas agree away from 3, but the left expression excludes 3. The unrestricted function \(y=x+2\) contains \((3,5)\), while the original rational function has a hole there.
2. Choose the Form That Reveals the Feature
| Representation | Typical structure | Information revealed quickly |
|---|
| Factored form | \(a(x-r_1)^{m_1}\cdots(x-r_k)^{m_k}\) | Real zeros, multiplicities, x-intercepts, denominator restrictions, holes, and vertical asymptotes |
| Standard form | \(a_nx^n+\cdots+a_1x+a_0\) | Degree, leading term, leading coefficient, y-intercept, and polynomial end behavior |
| Quotient-remainder form | \(q(x)+r(x)/g(x)\) | Slant or polynomial asymptote and remainder behavior |
| Expanded binomial form | \(\sum\binom{n}{k}a^{n-k}b^k\) | Coefficients and individual terms of a repeated binomial product |
No single form is always best. Rewrite with a purpose, then state what the new form reveals.
3. Polynomial Example: Factored and Standard Forms
\[P(x)=(x-2)^2(x+3).\]
Factored form immediately gives the zero \(x=2\) with multiplicity 2 and the zero \(x=-3\) with multiplicity 1. Expanding gives
\[P(x)=(x^2-4x+4)(x+3)=x^3-x^2-8x+12.\]
| From factored form | The graph touches at \(x=2\) and crosses at \(x=-3\). |
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| From standard form | The degree is 3, the leading coefficient is 1, and the y-intercept is 12. |
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| From both forms | The left tail falls, the right tail rises, and all intercept information is available. |
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4. Rational Example: Simplify Without Losing the Domain
\[R(x)=\frac{x^2-4}{x^2-x-2}=\frac{(x-2)(x+2)}{(x-2)(x+1)}.\]
The original denominator excludes \(x=2\) and \(x=-1\). After cancellation,
\[R(x)=\frac{x+2}{x+1},\qquad x\ne2,-1.\]
- Zero and x-intercept: \(x=-2\), so \((-2,0)\).
- Hole: substitute 2 into the simplified rule to get \((2,4/3)\).
- Vertical asymptote: \(x=-1\).
- Horizontal asymptote: equal leading degrees give \(y=1\).
5. Cancel Factors, Not Terms
Cancellation uses a common multiplicative factor:
\[\frac{(x+4)(x-1)}{(x+4)(x+2)}=\frac{x-1}{x+2},\qquad x\ne-4,-2.\]
It is invalid to cancel the \(x\) terms in \((x+4)/x\) because \(x+4\) is a sum, not a product containing a factor of \(x\).
6. The Polynomial Division Identity
If polynomial \(f\) is divided by nonzero polynomial \(g\), then
\[f(x)=g(x)q(x)+r(x),\qquad\deg r<\deg g.\]
Here \(q\) is the quotient and \(r\) is the remainder. Dividing both sides by \(g(x)\) gives
\[\frac{f(x)}{g(x)}=q(x)+\frac{r(x)}{g(x)},\qquad g(x)\ne0.\]
7. Complete Long-Division Example
Divide \(2x^3+3x^2-5x+7\) by \(x+2\).
| Step | Quotient term | Multiply and subtract |
|---|
| \(2x^3/x\) | \(2x^2\) | Subtract \(2x^3+4x^2\), leaving \(-x^2-5x+7\). |
| \(-x^2/x\) | \(-x\) | Subtract \(-x^2-2x\), leaving \(-3x+7\). |
| \(-3x/x\) | \(-3\) | Subtract \(-3x-6\), leaving remainder 13. |
\[\frac{2x^3+3x^2-5x+7}{x+2}=2x^2-x-3+\frac{13}{x+2},\qquad x\ne-2.\]
8. Include Zero Placeholders for Missing Powers
Before dividing \(x^4-5x^2+4\) by \(x-2\), write
\[x^4+0x^3-5x^2+0x+4.\]
The placeholders keep like powers aligned. Long division gives
\[x^4-5x^2+4=(x-2)(x^3+2x^2-x-2).\]
The remainder is 0, so \(x-2\) is a factor.
9. Quotient Form Reveals Asymptotes
In the complete division example,
\[\frac{2x^3+3x^2-5x+7}{x+2}=2x^2-x-3+\frac{13}{x+2}.\]
As \(x\to\pm\infty\), the remainder fraction approaches 0. Therefore the rational graph approaches the polynomial
\[y=2x^2-x-3.\]
If the quotient is linear, its graph is a slant asymptote. If the quotient has higher degree, it is a polynomial asymptote.
10. Verify Every Division Result
Multiply the divisor and quotient, then add the remainder:
\[(x+2)(2x^2-x-3)+13=2x^3+3x^2-5x+7.\]
Also check that the remainder degree is less than the divisor degree. These two checks catch sign and subtraction errors.
11. The Binomial Theorem
The binomial theorem expands a repeated product without multiplying one pair at a time:
\[(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k.\]
- The coefficients \(\binom{n}{k}\) come from row \(n\) of Pascal's Triangle.
- The exponent of \(a\) decreases from \(n\) to 0.
- The exponent of \(b\) increases from 0 to \(n\).
- Every term has total exponent \(n\).
12. Pascal's Triangle and a Worked Expansion
| Power \(n\) | Coefficient row |
|---|
| 0 | 1 |
| 1 | 1, 1 |
| 2 | 1, 2, 1 |
| 3 | 1, 3, 3, 1 |
| 4 | 1, 4, 6, 4, 1 |
| 5 | 1, 5, 10, 10, 5, 1 |
Using row 5 with \(a=x\) and \(b=-2\),
\[(x-2)^5=x^5-10x^4+40x^3-80x^2+80x-32.\]
The alternating signs come from successive powers of \(-2\), not from changing the Pascal coefficients.
13. Representation Choice in Context
- Use factored form to identify when a modeled quantity becomes zero or undefined.
- Use standard form to identify dominant growth and a starting value such as \(P(0)\).
- Use quotient-remainder form to separate a long-term trend from a vanishing correction term.
- Use an expanded binomial when individual coefficients represent contributions to a model.
After rewriting, interpret the extracted value in the stated domain and units.
14. Technology, Common Errors, and AP Reasoning
- Use a computer algebra system to verify expansion, factoring, and division, not to replace the written reasoning.
- Do not cancel terms across addition or subtraction.
- Do not drop original denominator restrictions after cancellation.
- Insert zero coefficients for missing powers during long division.
- Subtract every term of each intermediate product.
- Require \(\deg r<\deg g\) before stopping division.
- Do not omit lower quotient terms from an asymptote equation.
- Track the sign of \(b\) carefully in \((a+b)^n\).
Complete response pattern: show the equivalent form, preserve its restrictions, identify the newly visible feature, and explain why that feature answers the question.