Target Term Worked Solutions
Precalculus & Algebra II • Exemplar Guide
Target Term Worked Solutions
Binomial Theorem
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Mastering systematic 4-step solutions to extract requested coefficients without full expansion.
General Term Formula for \((a + b)^n\)
\(T_{r+1} = \binom{n}{r} a^{n-r} b^r\) (where \(r \in \{0, 1, \dots, n\}\))
Step 1: Components ➔ Step 2: Exponent Eq ➔ Step 3: Solve \(r\) ➔ Step 4: Evaluate
Problem 1 Standard Positive Terms
Find the coefficient of \(x^4\) in \((2x + 3)^7\)
1 Identify Terms
\(a = 2x\)
\(b = 3\)
\(n = 7\)
2 Set General Term
\(T_{r+1} = \binom{7}{r} (2x)^{7-r} (3)^r\)
\(= \binom{7}{r} 2^{7-r} 3^r \cdot x^{7-r}\)
3 Equate & Solve \(r\)
Target power is \(x^4\):
\(7 - r = 4\)
\(\mathbf{r = 3}\) (4th term)
4 Calculate Coefficient
\(\binom{7}{3} \cdot 2^4 \cdot 3^3\)
\(= 35 \cdot 16 \cdot 27\)
Answer: \(15,120\)
Problem 2 Handling Negative Signs Carefully
Find the coefficient of \(x^3\) in \((3x - 2)^6\)
1 Identify Terms
\(a = 3x\)
\(b = \mathbf{-2}\) (keep sign!)
\(n = 6\)
2 General Term & Sign
\(T_{r+1} = \binom{6}{r} (3x)^{6-r} (-2)^r\)
\(= \binom{6}{r} 3^{6-r} (-2)^r x^{6-r}\)
3 Target Exponent
Need power \(x^3\):
\(6 - r = 3\)
\(\mathbf{r = 3}\) (odd power)
4 Calculate Coefficient
\(\binom{6}{3} \cdot 3^3 \cdot (-2)^3\)
\(= 20 \cdot 27 \cdot (-8)\)
Answer: \(-4,320\)
Common Pitfall: Forgetting the negative sign on \(b = -2\). Since \(r = 3\) is odd, \((-2)^3 = -8\), producing a negative coefficient.
Problem 3 Rational Exponents & Reciprocal Powers
Find the coefficient of \(x^5\) in \(\left(x^2 - \frac{3}{x}\right)^7\)
1 Rewrite with Exponents
\(a = x^2\)
\(b = -3x^{-1}\)
\(n = 7\)
2 Combine Exponents of \(x\)
\((x^2)^{7-r} \cdot (x^{-1})^r\)
\(= x^{14-2r} \cdot x^{-r}\)
\(= x^{14 - 3r}\)
3 Equate to Target
Need power \(x^5\):
\(14 - 3r = 5\)
\(3r = 9 \implies \mathbf{r = 3}\)
4 Calculate Coefficient
\(\binom{7}{3} \cdot (-3)^3\)
\(= 35 \cdot (-27)\)
Answer: \(-945\)
Check: The full term is \(T_4 = 35 \cdot (x^2)^4 \cdot (-3/x)^3 = 35 \cdot x^8 \cdot (-27/x^3) = -945x^5\). Requested coefficient is \(-945\).
Advanced Techniques
Constant Terms & Compound Products
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Problem 4 The "Constant Term" (Independent of \(x\))
Find the term independent of \(x\) in \(\left(2x^2 - \frac{1}{x}\right)^6\)
1 Identify & General Term
\(a = 2x^2\), \(b = -x^{-1}\), \(n = 6\)
\(T_{r+1} = \binom{6}{r} (2x^2)^{6-r} (-x^{-1})^r\)
\(= \binom{6}{r} 2^{6-r} (-1)^r x^{12-2r-r}\)
2 Set Exponent = 0
"Independent of \(x\)" means \(x^0\):
\(12 - 3r = 0\)
\(3r = 12 \implies \mathbf{r = 4}\)
3 Evaluate Terms
\(\binom{6}{4} = \binom{6}{2} = 15\)
\(2^{6-4} = 2^2 = 4\)
\((-1)^4 = 1\) (even power)
4 Final Value
\(T_5 = 15 \cdot 4 \cdot 1\)
Constant Term: \(60\)
Problem 5 Polynomial Multiplied by a Binomial
Find the coefficient of \(x^3\) in \((2 - x)(1 + 3x)^5\)
Strategy: Distribute \((2 - x)\) across the expansion of \((1 + 3x)^5\). Two distinct combinations form \(x^3\):
\(2 \cdot [x^3\text{ term}] - x \cdot [x^2\text{ term}]\)
Contribution 1: from \(2 \cdot (x^3\text{ term})\) r = 3
In \((1 + 3x)^5\), term with \(x^3\) is \(\binom{5}{3} (1)^2 (3x)^3\):
\(\binom{5}{3} \cdot 3^3 = 10 \cdot 27 = 270x^3\)
Multiply by factor \(2\): \(2 \times 270 = \mathbf{540x^3}\)
Contribution 2: from \(-x \cdot (x^2\text{ term})\) r = 2
In \((1 + 3x)^5\), term with \(x^2\) is \(\binom{5}{2} (1)^3 (3x)^2\):
\(\binom{5}{2} \cdot 3^2 = 10 \cdot 9 = 90x^2\)
Multiply by factor \(-x\): \(-1 \times 90 = \mathbf{-90x^3}\)
Combine Total: Total coefficient of \(x^3 = 540 + (-90) = 450\)
Answer: \(450\)
Problem 6 Trinomial Expansion by Grouping
Find the coefficient of \(x^2\) in \((1 - 2x + 3x^2)^4\)
1. Group as Binomial
Write \([1 + (-2x + 3x^2)]^4\).
Expand with variable \(u = (-2x + 3x^2)\):
\(1 + 4u + 6u^2 + 4u^3 + u^4\).
Powers \(u^3\) & \(u^4\) produce only \(x^3\) or higher.
2. Extract \(x^2\) Contributions
• From \(4u = 4(-2x + 3x^2)\):
\(4 \times (3x^2) = \mathbf{12x^2}\)
• From \(6u^2 = 6(-2x + 3x^2)^2\):
\(6 \times [(-2x)^2] = 6(4x^2) = \mathbf{24x^2}\)
3. Sum Coefficients
Sum all \(x^2\) terms:
\(12x^2 + 24x^2 = 36x^2\)
Answer: \(36\)
Exam Verification Checklist: 1. Ensure \(r\) is an integer with \(0 \le r \le n\). 2. Keep negative signs with \(b\). 3. Always distinguish "term" (includes \(x^k\)) from "coefficient" (number only).
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