Find one term without expanding everything: Sometimes a question asks for just one term, such as the term, rather than the whole expansion. The examples below show how to find that term and how to give its coefficient.
Find one term
Find the term in of .
Step by step
- Read the two numbers for the binomial coefficient. In , the bracket power is . We want , so use at the bottom. The result, , is the binomial coefficient.
- Include the other part of the bracket. The power on is . Multiply the binomial coefficient by . This gives , the coefficient of .
- Write the whole term. A term includes its power of . Attach to the coefficient .
Final answer
The term in is .
- the power on the bracket
- pick r to land on the term you want (0, 1, 2, …)
Term or coefficient?: Binomial coefficient: .
Coefficient of : . Whole term: .
Find a coefficient (with a sign)
Find the coefficient of in .
Step by step
- Put over . is the bracket power; is the power of we want.
- Give the part power . The other part gets power . Raise each whole part, including its sign.
- Multiply the numbers. The coefficient is the number multiplying the power of .
Final answer
The coefficient of is .
When this shortcut works: Use this direct method when one part is a constant and the other is or a multiple of , such as . With or , first work out the resulting power of .
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What does independent of x mean?: Independent of means the power of is . Since (for ), a term such as becomes . Only a number remains: this is the constant term. In , the constant term is ; its is usually not written.
Start simple: no fractions
Find the term independent of in .
Step by step
- Expand the bracket. Squaring means multiplying two identical brackets.
- Look at each term. and contain . The last term, , does not.
- Connect this to the power rule. For non-zero , , so . This is why we look for power when finding a constant term.
Final answer
The term independent of is .
Now try a fraction: With a fraction such as , both parts contain . The constant term appears when the powers from the numerator and denominator cancel. Instead of expanding every term, we find the one whose total power of is . Here because is in a denominator.
The constant term
Find the term independent of in .
Step by step
- Choose the two parts. Here , and . In the general term, counts how many times we choose ; the other choices are .
- Separate the numbers from the powers of . Raise every factor to its power. An in the denominator gives a negative power: .
- Combine only the powers of . When multiplying powers with the same base, add their exponents. This is where the comes from.
- Make the power zero. We need no left, so solve . This gives : choose two parts and two parts. Since the term number is , this is the third term.
- Substitute into the whole term. The coefficient counts the ways to choose the two parts from four brackets.
- Cancel the factors and multiply the numbers. , so only a number remains.
Final answer
The term independent of is .
Check the pattern: The powers of across the five terms are . The third term is the only one with power . We found it without expanding all five terms.
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Work backwards to a constant: To find an unknown constant, write the given coefficient with the general term, set it equal, and solve. Even powers give two answers.
Find the constant a
In the coefficient of is . Find .
Step by step
- Which term? Power of is , so needs . Coefficient is .
- Solve: .
Final answer
(both, unless the question says ).
Two answers from an even power: An even power of the constant gives two values — state both unless the question rules one out.
A coefficient inside too
In the coefficient of is . Find .
Step by step
- Which term? Power of is , so needs . Coefficient is .
- Solve: .
Final answer
.
Relationships between terms: Mean means average. If the third term is the mean of the second and fourth, add those two terms and divide by : . So .
Use the whole terms, including . For example, use , not just .
Mean of three binomial terms
In the expansion of in descending powers of , the third term is the mean of the second and fourth terms. Given , find the possible values of .
Step by step
- Identify the whole terms. Use for the second, third and fourth terms.
- Translate mean into an equation. Twice the middle equals the sum of the outer terms.
- Use . Divide by and rearrange.
- Solve the quadratic. Use the quadratic formula and keep both non-zero roots.
Final answer
.
Order and restrictions: Check whether the expansion is in ascending or descending powers. The term numbers change with the order. Only divide by a power of after checking that ; do not include an excluded zero root.
Find the power n: When the power is unknown, use the simplest coefficient to build an equation in and solve for the positive whole number.
One unknown power
In the coefficient of is . Find .
Step by step
- The coefficient is . Set it to 36.
- Clear and factorise. , so , giving .
Final answer
(reject ).
Keep the whole-number root: Only a positive whole number can be a power, so discard any negative or fractional root.
Two unknowns, two coefficients
In the expansion of , the coefficient of is and the coefficient of is . Find and .
Step by step
- Two equations. The coefficient of is , and the coefficient of is . So and .
- Substitute into the second: tidies to , so .
Final answer
and .
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An exam-style question
Find the coefficient of in the expansion of .
Step by step
- Put over . is the bracket power; is the power of we want.
- Give the part power . The other part gets power . Raise each whole part, including its sign.
- Multiply the numbers. The coefficient is the number multiplying the power of .
Final answer
The coefficient of is .
To get full marks
- Show the selected term: .
- Keep the sign: .
- Give the coefficient , not the whole term .