Finding a Specific Term Practice Test

Advanced Algebra Practice Test: ACT math skills.

Finding a Specific Term Practice Test

This test has 20 questions

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Specific Terms · Coordinate Guide

Locate one term without expanding them all.

A specific-term problem is an indexing problem first and an arithmetic problem second. Translate the requested position into the theorem index, preserve each complete base, and calculate only the term that reaches the target coordinate.

This high-school review covers term position, the general term, terms counted from either end, target powers, coefficients, constant terms, middle terms, negative signs, and fast structural checks.

PositionRead the requested place carefully.

IndexConvert position before substitution.

CoefficientSelect one Pascal or combination value.

PowersTrack both complete bases.

CheckConfirm degree, sign, and location.

Map 01

The general term is the locator formula

It identifies any single term in a binomial expansion without writing the entire polynomial.

Master coordinate

T_{r + 1} = (\binom{n}{r}) a^{n - r} b^{r}

ExpansionThe outside exponent determines the coefficient row.

Term indexThe index starts at zero even though positions start at one.

First baseIts exponent decreases as the route moves right.

Second baseIts exponent increases from zero.

Essential conversion: if the question asks for the term in position

k

, use

r = k - 1

. The first term therefore uses an index of zero, not one.

Map 02

Position and theorem index differ by one

Mark the requested position before touching coefficients or exponents.

Index conversion

Read the subscript literally

The formula labels its term with one more than the index. That is why the fourth term uses an index of three.

k = 4 \to r = 4 - 1 = 3

Make this conversion as a separate line. It prevents the most common off-by-one error.

Map 03

Worked route: find the fourth term

Substitute only after the position has been converted to its index.

Single-term route

{(x + 2)}^{6}

Position map

r = 4 - 1 = 3

The fourth term uses the third index.

Substitute

T_{4} = (\binom{6}{3}) x^{6 - 3} 2^{3}

Both power tracks are now fixed.

Evaluate

20 \cdot x^{3} \cdot 8

Evaluate the combination and numerical power.

Located term

T_{4} = 160 x^{3}

Report the full term, including its variable part.

Map 04

Negative bases keep their sign inside the power

The sign of a selected term depends on the exponent applied to the negative base.

Sign terrain

Target

Find the fifth term of the expansion.

{(2 x - 3)}^{7}

r = 5 - 1 = 4

Substitution

T_{5} = (\binom{7}{4}) {(2 x)}^{3} {(- 3)}^{4}

T_{5} = 22680 x^{3}

The fourth power makes the negative base positive.

Sign checkpoint: do not alternate signs by memory alone. Keep the negative base grouped, then decide whether its selected exponent is odd or even.

Map 05

A requested variable power determines the index

Build and solve the exponent equation before evaluating the coefficient.

Power coordinate

Worked target-power example

Find the term containing $x^{4}$ .

{(3 x^{2} + 2)}^{4}

2 (4 - r) = 4 \to r = 2

Selected term

(\binom{4}{2}) {(3 x^{2})}^{2} 2^{2}

Use the index found from the power equation.

Result

6 \cdot 9 x^{4} \cdot 4 = 216 x^{4}

The complete requested term includes its variable factor.

Map 06

The constant term has a net variable exponent of zero

Positive and negative exponent contributions must balance.

Zero-power location

Expansion

{(x^{2} + \frac{2}{x})}^{6}

The first base contributes two variable powers each time; the second contributes one negative variable power.

Exponent equation

2 (6 - r) - r = 0

12 - 3 r = 0 \to r = 4

Coefficient

(\binom{6}{4}) 2^{4} = 15 \cdot 16

The variable factor equals one at zero power.

Constant term

240

Report a number because no variable remains.

Map 07

Terms counted from the end use a reversed position map

Translate the wording into a start position or directly into the matching index.

Reverse route

r = n - k + 1

Example request

Find the third term from the end.

{(p + 2 q)}^{6}

r = 6 - 3 + 1 = 4

Selected term

(\binom{6}{4}) p^{2} {(2 q)}^{4}

240 p^{2} q^{4}

There are seven total terms, so this is also the fifth term from the start.

Map 08

Middle-term questions begin with the total term count

An odd number of terms has one middle; an even number has two.

Center coordinates

One middle term

An even outside exponent creates an odd total number of terms.

{(a + b)}^{8}

T_{5} = 70 a^{4} b^{4}

Two middle terms

An odd outside exponent creates an even total number of terms.

{(x + y)}^{5}

T_{3} = 10 x^{3} y^{2}, T_{4} = 10 x^{2} y^{3}

Count first: a power of

n

produces

n + 1

ordinary terms before any accidental cancellation. Use that count to locate the center.

Map 09

A coefficient request is not the same as a term request

Locate the full term, then report only the numerical factor if the wording asks for a coefficient.

Output boundary

{(2 x - 1)}^{7}

Match the power

7 - r = 5 \to r = 2

Full term

672 x^{5}

Requested coefficient

672

Map 10

Structural checks verify a located term quickly

These checks catch index, exponent, coefficient, and sign errors without a full expansion.

Route validation

1Position checkThe selected index must be exactly one less than the start position.

2Index rangeThe index must be a whole number from zero through the outside exponent.

3Exponent sumThe two binomial exponents must add to the outside exponent.

4Complete-base checkNumerical coefficients and signs stay inside their selected powers.

5Parity checkAn odd power preserves a negative sign; an even power removes it.

6Requested outputDistinguish a full term, a coefficient, a position, and a constant.

(n - r) + r = n

Map 11

Skills Covered

The test measures accurate navigation from wording to one selected term.

Map legend

Use the general termSubstitute one index without expanding the full power.

Convert position to indexAvoid shifting every selected term by one place.

Match a target powerSolve an exponent equation to locate the correct index.

Find constant termsSet the net variable exponent equal to zero.

Locate middle and reverse termsUse total term count and direction carefully.

Report the requested objectSeparate a complete term from its numerical coefficient.

Map 12

How to Approach the Test

Use a fixed route so that arithmetic begins only after the location is secure.

Six-stop route

Identify the requested output

Decide whether the question asks for a position, term, coefficient, constant, or middle term.

Underline the output type.

Mark both complete bases

Keep numerical factors, variables, fractions, and negative signs grouped.

Do not simplify bases early.

Determine the index

Convert an ordinal position or solve the required exponent equation.

Check that the index is valid.

Substitute into one term

Write the combination coefficient and both exponent tracks.

Exponents must sum correctly.

Evaluate signs and numbers

Use odd-even reasoning before multiplying large coefficients.

Keep exact integer arithmetic.

Match the requested form

Return the full term or remove the variable part when only a coefficient is requested.

Finish with a structural check.

Map 13

Common Mistakes

Most wrong answers come from choosing the wrong coordinate before calculating.

Wrong turns

Using the position as the index

The first position corresponds to an index of zero, so subtract one first.

Expanding the entire polynomial

This adds unnecessary arithmetic and more chances for sign errors.

Reversing the two power tracks

The first exponent decreases while the second exponent increases.

Dropping part of a base

A coefficient or negative sign inside parentheses receives the exponent too.

Guessing an alternating sign

Evaluate the selected power of the negative base directly.

Counting backward incorrectly

Use total terms or the reverse-index formula before substitution.

Reporting a coefficient as a term

A complete term includes its variable factor unless the prompt says otherwise.

Accepting an impossible index

A negative, fractional, or oversized index means the requested term does not occur.

Final map

Specific-term audit

Confirm the destination, route, selected coefficient, exponents, sign, and reported form.

Location confirmed

Term Coordinate Check

A correct answer follows one precise path from the wording to a valid index and then to one fully evaluated term.

Locate · Substitute · Evaluate · Verify

Did I identify exactly what must be reported?Separate position, term, coefficient, and constant-term requests.

Did I convert the requested position correctly?Start positions begin at one; theorem indices begin at zero.

Is the selected index a valid whole number?It must lie within the available expansion range.

Did I preserve both complete bases?Include coefficients, variables, fractions, and negative signs.

Do the powers and sign match the selected index?Check complementary exponents and odd-even behavior.

Does the final form answer the wording?Report the full term unless only its coefficient is requested.

Use this free 20-question practice test for high school Advanced Algebra review, ACT-style skill practice, placement preparation, or classroom practice. You can retake the test without creating an account. The examples in this review block are illustrative and are not copies of the test questions.