What a question asks: A question wants just one term — like the term of , or a coefficient. The quick way: the choose-number's top is the bracket power, and its bottom is the power of you want. Then fill in the powers. Two examples:
Find one term
Find the term in of .
Step by step
- Read the choose straight off. Top = the bracket power 6, bottom = the power of you want, 3. So it's .
- Fill in the powers. to the power 3, and the fills the other . Multiply:
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?: If a term is asked for, keep the power of ; if a coefficient is asked for, give the number only. And watch signs: an odd power of a negative term stays negative.
Find a coefficient (with a sign)
Find the coefficient of in .
Step by step
- Read the choose straight off. Top = the bracket power 4, bottom = the power of , 3. So it's . (The sits inside , so is used 3 times.)
- Fill in the powers. , and the fills the other : . Multiply:
Final answer
The coefficient of is .
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The term with no x: The term independent of (the constant term) is the one where the power of is . Set the power to , solve for , then work out that single term.
The constant term
Find the term independent of in .
Step by step
- General term with , . The lowers the power, so the power of is .
- Set the power to 0: , so . Then and .
Final answer
The term independent of is .
Mind the 1/x: A makes the power of fall as rises, so here it is , not . Always solve the power equation for first.
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Work backwards to a constant: Given a coefficient's value, write it with the general term, set it equal, and solve for the unknown. An even power can give two answers ().
Find the constant k
In the coefficient of is . Find .
Step by step
- Which term? The power of is , so needs . Its coefficient is .
- Set equal to 40 and solve. , so .
Final answer
(both, unless the question says ).
Two answers from an even power: An even power of means can be positive or negative — give both values unless the question restricts it.
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
.
Find the power n: When the power is unknown, use the simplest coefficient to make 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 . Write it as and set it to 28.
- Clear and factorise. , so , giving .
Final answer
(reject : the power must be a positive whole number).
Keep the whole-number root: must be a positive whole number, so throw away 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
- Write the general term with , . The power of is .
- Match the power: , so . Then , , .
- Multiply, keeping the sign. .
Final answer
The coefficient of is .
To get full marks
- Write the general term and match the power of to 2 (so ).
- Work out , and , keeping the minus sign.
- Multiply to get the coefficient and state it.