HSA.APR.C.5: The Binomial Theorem and Pascal's Triangle
In plain English: HSA.APR.C.5 is an advanced (+) Common Core algebra standard that asks students to know and apply the Binomial Theorem: (x + y)ⁿ expands into n + 1 terms whose coefficients come from row n of Pascal's Triangle or from the combinations C(n, k). Because x and y can be any numbers, it also expands (2x - 3)⁴ or 1.01⁵. It is usually taught in Algebra II or Precalculus.
(+) Know and apply the Binomial Theorem for the expansion of (x + y)ⁿ in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.
Common Core State Standards for Mathematics · Domain: Arithmetic with Polynomials and Rational Expressions (APR) · Cluster: Use polynomial identities to solve problems Also written as HSA-APR.C.5 or A-APR.5 · Official standard
Students build Pascal's Triangle, see why its rows are the coefficients of (x + y)ⁿ, and state the Binomial Theorem: (x + y)ⁿ is a sum of n + 1 terms C(n, k)xn-kyk, where the exponents in each term add to n. A tree of choices (x or y from each factor) explains why the coefficient C(n, k) counts the ways to pick y from k of the n factors.
Because the theorem holds for any numbers x and y, students then apply it where x and y are not single letters: (2x - 3)³, (x² - 2)³, 1.01⁵ written as (1 + 0.01)⁵, and (1 + √2)³. They also learn to find a single term, such as the x³ term of (x + 2)⁷, without writing the whole expansion. This is a (+) standard: Common Core lists it as additional mathematics for students who take advanced courses.
Learning Objectives
By the end of this lesson, students will be able to:
Build rows of Pascal's Triangle and explain why each entry is the sum of the two entries above it
State the Binomial Theorem for (x + y)ⁿ and describe the number of terms and the pattern of exponents
Find binomial coefficients with Pascal's Triangle or with C(n, k) = n!/(k!(n - k)!)
Expand powers of binomials in which x and y stand for any numbers or expressions, including negative terms, coefficients and radicals
Find a single term or coefficient of an expansion without expanding completely
Prior Knowledge Required
Students should already be comfortable with:
Multiplying polynomials HSA.APR.A.1
Properties of integer exponents, such as (2x)³ = 8x³ 8.EE.A.1
Proving and using polynomial identities such as (a + b)² = a² + 2ab + b² HSA.APR.C.4
Factorial notation and counting combinations, helpful but not required
Pairs multiply out two powers by hand and write only the coefficients in a row.
Warm-Up Prompt
"Expand (x + y)² and (x + y)³ by multiplying. List the coefficients of each result in order. Without multiplying, predict the coefficients of (x + y)⁴, then check by multiplying (x + y)³ by (x + y)."
Record the rows 1 2 1 and 1 3 3 1 on the board. Many students predict 1 4 4 1 or 1 4 8 4 1 for the fourth power. Multiplying (x + y)(x³ + 3x²y + 3xy² + y³) shows why the answer is 1 4 6 4 1: each new coefficient is the sum of two coefficients from the row before. This sets up Pascal's Triangle.
Direct Instruction20-25 minutes
Pascal's Triangle and the theorem. Build rows 0 to 6 with the class (Diagram 1). Then state the Binomial Theorem: for a positive integer n and any numbers x and y, (x + y)ⁿ = C(n, 0)xⁿ + C(n, 1)xn-1y + C(n, 2)xn-2y² + ... + C(n, n)yⁿ, where C(n, k) = n!/(k!(n - k)!) is entry k of row n (counting from 0). Use Diagram 2 to explain the coefficients: each term comes from choosing x or y from every factor, and C(n, k) counts the ways to choose y from exactly k factors. To expand any binomial power, follow these steps:
Identify the two terms: in (2x - 3)³, the first term is 2x and the second term is -3, sign included.
Write the coefficients from row n of Pascal's Triangle, or compute C(n, k).
Attach powers: the first term's power falls from n to 0 while the second term's power rises from 0 to n.
Simplify each term: raise every coefficient and sign inside the parentheses to its power.
Check: there are n + 1 terms, and substituting x = 1 into both forms gives the same number.
Expand (1 + √2)³ with row 3, using (√2)² = 2 and (√2)³ = 2√2.
Equation: 1 + 3√2 + 3 · 2 + 2√2 = 7 + 5√2
After the fourth example, point out that the terms shrink quickly when y is small, which is why the first few terms of (1 + 0.01)ⁿ give a good estimate. After the fifth, stress what "x and y are any numbers" means: the theorem is an identity, so it holds whatever is substituted for x and y.
Guided Practice15 minutes
Work through three tasks with the class, one student at the board per step. (1) Build row 5 from row 4 and expand (x + y)⁵ = x⁵ + 5x⁴y + 10x³y² + 10x²y³ + 5xy⁴ + y⁵. (2) Expand (a - 2)⁴ = a⁴ - 8a³ + 24a² - 32a + 16, writing the powers of -2 (1, -2, 4, -8, 16) in a row first. (3) Find the coefficient of x²y⁴ in (x + y)⁶ using C(6, 4) = 15, then confirm it in row 6. Watch for these errors: writing (2x)³ as 2x³, losing the signs of a negative second term, and exponents that do not add to n.
Independent Practice15 minutes
Students complete four tasks on their own: (1) expand (3x + 1)³ = 27x³ + 27x² + 9x + 1; (2) expand (x - y)⁵ = x⁵ - 5x⁴y + 10x³y² - 10x²y³ + 5xy⁴ - y⁵ and describe the sign pattern; (3) find the fourth term of (x + 2)⁶, which is C(6, 3)x³2³ = 160x³; (4) find the sum of the coefficients of (x + y)⁸ by substituting x = y = 1 (2⁸ = 256). Students check tasks 1 and 2 by substituting x = 1 (and y = 1) into both forms.
Closure5-10 minutes
Exit ticket: (1) Row 6 is 1 6 15 20 15 6 1. Write row 7. (1 7 21 35 35 21 7 1.) (2) Expand (x + 1)⁴. (x⁴ + 4x³ + 6x² + 4x + 1.) (3) How many terms does (x + y)¹⁰ have, and what do the exponents in each term add to? (11 terms; the exponents add to 10.)
Differentiation Strategies
For Struggling Students
Give a printed Pascal's Triangle and a term template with blanks: ( )·( )^__·( )^__ for each term
Have students write the first term and the second term, sign included, in a two-row table of powers before multiplying
Start with (x + 1)ⁿ, where every power of the second term is 1, before moving to (x - 2)ⁿ
For Advanced Students
Prove that C(n, k - 1) + C(n, k) = C(n + 1, k) from the factorial formula, which explains the addition rule of Pascal's Triangle
Explain why (1 + 1/n)ⁿ is always less than 3 by comparing the terms of its expansion with 1 + 1 + 1/2 + 1/4 + ...
Use the theorem with x = 1 and y = -1 to show that the alternating sum of every row n ≥ 1 is 0
Assessment Guidance
What to Look For
Check that students treat the whole first term and the whole second term as the x and y of the theorem: (2x)³, not 2x³, and (-3)², not -3². Every expansion should have n + 1 terms with exponents that add to n. When students find a single term, ask which power belongs to which term and which C(n, k) goes with it. Students should be able to explain in words why the coefficients are the entries of Pascal's Triangle, using either the addition rule or the choosing argument.
02
Classroom Activities
3 Activities
1
Build and Explore Pascal's Triangle
20 minPairs
Pairs fill in rows 0 to 10 of a printed triangle template using the addition rule, then look for patterns and connect each one to the Binomial Theorem.
Procedure
Fill in rows 0 to 10. Check that row 10 has 11 entries and reads the same in both directions
Add each row and record the sums 1, 2, 4, 8, ... Explain the pattern by substituting x = y = 1 into (x + y)ⁿ
Compute the alternating sum of rows 1 to 6 (1 - 4 + 6 - 4 + 1 and so on). Explain the result with x = 1, y = -1
Multiply (x + y)(x³ + 3x²y + 3xy² + y³) and circle where each coefficient of row 4 comes from
Discussion Questions
Why is every row symmetric? What does swapping x and y do to (x + y)ⁿ?
Why does row n have n + 1 entries?
Which entry of row 10 is the coefficient of x³y⁷?
Modification for Distance Learning
Use a shared spreadsheet where each cell adds the two cells above it. Students type the formula once, fill it across, and compare the result with their hand-built rows.
2
Choosing x or y: Where the Coefficients Come From
20 minGroups of 3
Groups list every way to pick one letter from each factor of (x + y)(x + y)(x + y) and then (x + y)⁴, and discover that the coefficients count choices.
Procedure
For three factors, list all 8 strings (xxx, xxy, xyx, ...), as in Diagram 2, and group them by the number of y's: 1, 3, 3, 1
For four factors, list all 16 strings and group them: 1, 4, 6, 4, 1
Compare with C(4, k) for k = 0 to 4 and with row 4 of Pascal's Triangle
Without listing, predict how many strings of length 5 contain exactly two y's (C(5, 2) = 10)
Discussion Questions
Why does each string correspond to one term before like terms are combined?
How many strings are there for n factors, and how does that match the row sums?
Challenge Variation
A fair coin is tossed 5 times. Use row 5 to find how many of the 32 equally likely outcomes have exactly 3 heads (10), and explain the link to the coefficient of x²y³ in (x + y)⁵.
3
Expansion Relay
20 minPairs
Each pair gets 6 cards. Partner A writes the coefficients and the powers of each term; Partner B simplifies and checks by substituting a value. Partners switch roles on every card.
Cards are face down; pairs flip one at a time and race the clock (3 minutes per card)
Partner B checks each answer by substituting x = 1 (or a = 1, b = 1) into both forms
Pairs swap card sets with another pair and check each other's work against the answer key
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Diagrams & Visual Aids
2 diagrams
Diagram 1: Pascal's Triangle and the Addition Rule
Rows 0 to 6 of Pascal's Triangle. Each entry is the sum of the two entries above it, and row n gives the coefficients of (x + y)ⁿ. The row sums are powers of 2 because (1 + 1)ⁿ = 2ⁿ.
Diagram 2: Why the Coefficients Count Choices
Multiplying out (x + y)³ means choosing x or y from each of the three factors, which gives 2³ = 8 strings. The coefficient of x²y is 3 because exactly 3 strings contain one y: C(3, 1) = 3.
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Homework Assignment
~30 min
HSA.APR.C.5 Homework: Expanding with the Binomial Theorem
Directions: Show the coefficient, the power of the first term and the power of the second term for every term you write. Check each full expansion by substituting a value such as x = 1 into both forms. Leave answers exact unless a problem asks for a decimal.
Part 1: Coefficients and Full Expansions (Problems 1-3)
Write rows 0 through 8 of Pascal's Triangle. Use row 8 to write the full expansion of (a + b)⁸, and check that the sum of the coefficients is 2⁸.
Expand (x - 4)⁴. Show the powers of -4 that you use.
Expand (3x + 2y)³ and simplify every term.
Part 2: Single Terms and Numbers (Problems 4-6)
(a) Find the term containing x⁵ in the expansion of (x - 3)⁸. (b) Find the coefficient of x²y⁵ in the expansion of (2x + y)⁷. Show which C(n, k) you used in each part.
A savings account pays 2% interest once a year, so $500 grows to 500(1.02)⁶ dollars in 6 years. Write 1.02⁶ as (1 + 0.02)⁶ and use the first four terms of the expansion to estimate the balance to the nearest cent. Explain why the remaining terms do not change the answer to the nearest cent.
Expand (2 + √3)⁴ and write the result in the form p + q√3. Then, without expanding again, write (2 - √3)⁴ in the same form and explain your reasoning.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Coefficients
Correct row of Pascal's Triangle or C(n, k) for every term
One coefficient wrong
Coefficients missing or guessed
Powers and Signs
Whole terms raised to powers, signs handled, exponents add to n
One sign or power error
Several errors
Single Terms
Correct term with C(n, k) and powers shown
Correct method with an arithmetic error
Not attempted or wrong method
Numerical Applications
Correct estimate, rounding and explanation
Correct value without explanation
Missing or incorrect
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Quiz: 20 Questions
Interactive, with answers
Instructions
Answer each question, then read the explanation. The score counts your multiple-choice answers, and Reset quiz clears everything for a fresh attempt.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
What is the coefficient of x⁷y² in the expansion of (x + y)⁹?
Answer: A
The coefficient is C(9, 2) = 9 · 8/2 = 36, the third entry of row 9 (1, 9, 36, ...). Choice C forgets to divide by 2!. Choice D is C(9, 3), the coefficient of x⁶y³. Choice B is C(9, 1).
Question 2 of 20 · Multiple Choice
How many terms does the expansion of (x + y)¹² have?
Answer: C
The power of y runs from 0 to 12, which gives 13 terms. Choice A forgets the term with y⁰. Choices B and D confuse the number of terms with 2 · 12 and 12².
Question 3 of 20 · Multiple Choice
In every term of the expansion of (x + y)ⁿ, the exponents of x and y:
Answer: B
Each term comes from choosing x from some factors and y from the rest of the n factors, so the exponents add to n, as in x⁴y² for n = 6. Choice D confuses the exponent sum with the number of terms, n + 1.
Question 4 of 20 · Multiple Choice
What is the coefficient of x⁹y² in the expansion of (x + y)¹¹?
Answer: D
C(11, 2) = 11 · 10/2 = 55. Choice A forgets to divide by 2. Choice B multiplies 11 by the exponent 2. Choice C is C(11, 1), the coefficient of x¹⁰y.
Question 5 of 20 · Multiple Choice
Expand (x + 4)³.
Answer: A
Row 3 is 1 3 3 1: x³ + 3x²(4) + 3x(4²) + 4³ = x³ + 12x² + 48x + 64. Choice B leaves out the middle terms. Choice C uses the powers of 4 without the coefficients 3. Choice D uses 4 instead of 4² in the third term.
Question 6 of 20 · Multiple Choice
Expand (5 - x)⁴.
Answer: C
With first term 5, second term -x and row 4 (1 4 6 4 1): 5⁴ + 4(5³)(-x) + 6(5²)(-x)² + 4(5)(-x)³ + (-x)⁴ = 625 - 500x + 150x² - 20x³ + x⁴. The signs alternate because odd powers of -x are negative. Choice A ignores the minus sign. Choice B leaves out the coefficients from row 4. Choice D makes every middle term negative.
Question 7 of 20 · Multiple Choice
What is the coefficient of x² in the expansion of (3x - 1)⁴?
Answer: B
The x² term is C(4, 2)(3x)²(-1)² = 6 · 9x² · 1 = 54x². Choice A leaves out the factor 3² = 9. Choice C gets the sign wrong: (-1)² = 1. Choice D leaves out the coefficient C(4, 2) = 6.
Question 8 of 20 · Multiple Choice
Row 9 of Pascal's Triangle is 1 9 36 84 126 126 84 36 9 1. Which is row 10?
Answer: D
Add neighboring pairs of row 9: 1 + 9 = 10, 9 + 36 = 45, 36 + 84 = 120, 84 + 126 = 210, 126 + 126 = 252, then the row repeats by symmetry. Row 10 has 11 entries. Choice A has only 10 entries because it misses the middle sum 252. Choice C adds 36 + 9 incorrectly.
Question 9 of 20 · Multiple Choice
What is the sum of the coefficients of the expansion of (2x + y)⁴?
Answer: A
Substitute x = 1 and y = 1: (2 + 1)⁴ = 3⁴ = 81. Choice B is the row sum of row 4 (2⁴), which ignores the coefficient 2 on x. Choices C and D do not come from a valid substitution.
Question 10 of 20 · Multiple Choice
A student writes (x - 2)³ = x³ - 6x² - 12x - 8. What is the error?
Answer: B
The third term is 3x(-2)² = 3x · 4 = 12x, which is positive because (-2)² is positive. So (x - 2)³ = x³ - 6x² + 12x - 8. Choice C leaves out the middle terms. Choice D forgets to multiply the coefficient 3 by -2.
Question 11 of 20 · Multiple Choice
Which is the term containing y³ in the expansion of (x + 2y)⁵?
Answer: C
The term is C(5, 3)x²(2y)³ = 10 · x² · 8y³ = 80x²y³. Choice A leaves out the factor 2³ = 8. Choice B uses 2² instead of 2³. Choice D multiplies 10 by the exponent 3.
Question 12 of 20 · Multiple Choice
Use the Binomial Theorem with (1 - 0.01)⁴ to find 0.99⁴ exactly.
Answer: D
(1 - 0.01)⁴ = 1 - 4(0.01) + 6(0.0001) - 4(0.000001) + 0.00000001 = 0.96059601. Choice A keeps only the first two terms. Choice B is (1 + 0.01)⁴: it ignores the minus sign. Choice C is 0.98², not 0.99⁴ (keeping three terms would give 0.9606).
Question 13 of 20 · Multiple Choice
Which expression gives the coefficient of x⁵y³ in the expansion of (x + y)⁸?
Answer: B
The coefficient is C(8, 3) = 8!/(5! 3!) = 56, entry 3 of row 8. Choice C forgets to divide by 3!, so it counts ordered choices instead of combinations. Choices A and D multiply exponents, which has no meaning here.
Question 14 of 20 · Multiple Choice
Expand (x² - 2)³.
Answer: A
With first term x² and second term -2: (x²)³ + 3(x²)²(-2) + 3(x²)(-2)² + (-2)³ = x⁶ - 6x⁴ + 12x² - 8. Choice C gets the sign of the third term wrong: (-2)² = 4 is positive. Choice D leaves out the coefficients 3 from row 3.
Question 15 of 20 · Short Answer
Expand (2x + 1)⁵.
Row 5 is 1 5 10 10 5 1. The terms are (2x)⁵ + 5(2x)⁴ + 10(2x)³ + 10(2x)² + 5(2x) + 1 = 32x⁵ + 80x⁴ + 80x³ + 40x² + 10x + 1. Check with x = 1: 3⁵ = 243 and 32 + 80 + 80 + 40 + 10 + 1 = 243.
Question 16 of 20 · Short Answer
Explain why the coefficients of (x + y)ⁿ read the same from left to right and from right to left. Use (x + y)⁷ as an example.
The coefficient of xn-kyk counts the ways to choose y from k factors, which is the same as choosing x from the other n - k factors, so C(n, k) = C(n, n - k). Swapping x and y does not change (x + y)ⁿ, so the coefficient of x⁵y² equals the coefficient of x²y⁵. For n = 7: C(7, 2) = C(7, 5) = 21, and row 7 is 1 7 21 35 35 21 7 1.
Use the Binomial Theorem to explain why the entries of row n of Pascal's Triangle add to 2ⁿ. Then find the sum of row 11.
The entries are the coefficients of (x + y)ⁿ. Substituting x = 1 and y = 1 makes every term equal to its coefficient, and the left side becomes (1 + 1)ⁿ = 2ⁿ. So the sum of row 11 is 2¹¹ = 2048.
Question 20 of 20 · Short Answer
Find the coefficient of x⁴ in the expansion of (2x - 1)⁶.
The x⁴ term is C(6, 2)(2x)⁴(-1)² = 15 · 16x⁴ · 1 = 240x⁴, so the coefficient is 240.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSA.APR.C.5 mean?
HSA.APR.C.5 means students know the Binomial Theorem and can use it to expand (x + y)ⁿ for any positive integer n. The coefficients are the entries of row n of Pascal's Triangle, the exponents of x fall from n to 0 while those of y rise from 0 to n, and x and y may be any numbers or expressions.
What does the (+) in HSA.APR.C.5 mean?
The (+) marks an advanced standard. Common Core describes (+) standards as additional mathematics that students should learn in order to take advanced courses such as calculus or discrete mathematics. Many schools teach this standard in Algebra II or Precalculus, but it is not part of the core expected of every student.
Should students use Pascal's Triangle or the combination formula?
Use Pascal's Triangle for small powers and C(n, k) = n!/(k!(n - k)!) for large powers or single terms. Building row 12 by hand is slow, but C(12, 3) = 220 takes one line. Students should know that both give the same numbers and why.
What are common mistakes when expanding binomials?
A common mistake is raising only the variable to a power, writing (2x)³ as 2x³ instead of 8x³. Other frequent errors are dropping the sign of a negative second term, using the wrong row of the triangle (row 5 has six entries), and writing exponents that do not add to n.
How do you find one term of a binomial expansion without expanding everything?
Decide the power of each part, then use C(n, k). For the x³ term of (x - 2)⁶, the power of -2 must be 3, so the term is C(6, 3)x³(-2)³ = 20 · (-8)x³ = -160x³. The power k of the second term tells you which C(n, k) to use.
Why does the standard say x and y are "any numbers"?
The Binomial Theorem is an identity, so it holds whatever is substituted for x and y: integers, fractions, decimals, radicals, negative numbers or whole expressions like 2x or -3y². That is why the same pattern expands (2x - 3y)⁵ and gives 1.01⁵ = 1.0510100501 exactly.
How is the Binomial Theorem proved?
The Common Core notes that it can be proved by mathematical induction or by a combinatorial argument. The combinatorial argument is the one in Diagram 2: each term of the product comes from choosing x or y from each factor, and C(n, k) counts the choices with exactly k y's. Induction uses the addition rule of Pascal's Triangle.
Is HSA.APR.C.5 in Algebra 2 or Precalculus?
Both are common. Some Algebra II courses teach it at the end of the polynomial unit, right after polynomial identities (HSA.APR.C.4), and some Precalculus courses teach it with sequences, series and counting.
How is the Binomial Theorem connected to probability?
The same numbers C(n, k) count outcomes. When a fair coin is tossed 6 times, the number of outcomes with exactly 2 heads is C(6, 2) = 15 out of 2⁶ = 64. This link leads to combinations in probability (HSS.CP.B.9) and to binomial probability later on.
How can parents help a student practice the Binomial Theorem?
Ask your student to build Pascal's Triangle from memory using the addition rule, then to explain what row 4 has to do with (x + y)⁴. A good check question is "How do you know you have all the terms?" The answer: there are n + 1 terms and the exponents in every term add to n.
07
Related Standards
5 standards
These standards connect to HSA.APR.C.5: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
HSA.APR.A.1Prerequisite
Add, subtract and multiply polynomials; polynomials are closed under these operations
