HSA.APR.A.1: Adding, Subtracting and Multiplying Polynomials
In plain English: HSA.APR.A.1 is the Common Core algebra standard that asks students to add, subtract and multiply polynomials and to understand that polynomials form a system like the integers. The sum, difference or product of two polynomials is always another polynomial, while division does not always stay inside the system. It is usually taught in Algebra I and revisited in Algebra II.
Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Common Core State Standards for Mathematics · Domain: Arithmetic with Polynomials and Rational Expressions (APR) · Cluster: Perform arithmetic operations on polynomials Also written as HSA-APR.A.1 or A-APR.1 · Official standard
Students add, subtract and multiply polynomials, and they learn why the answer is always another polynomial. The lesson uses the comparison the standard names: polynomials behave like the integers. Adding, subtracting or multiplying two integers always gives an integer, and adding, subtracting or multiplying two polynomials always gives a polynomial. Dividing does not always stay inside either system.
The comparison is concrete, not only a slogan. When x = 10, a polynomial such as 2x + 3 is the integer 23, and multiplying (2x + 3)(x + 4) is the same work as multiplying 23 × 14 with place value. Students use this to check their algebra and to see that combining like terms plays the role of lining up place values.
Learning Objectives
By the end of this lesson, students will be able to:
Add and subtract polynomials by combining like terms, distributing a subtraction sign across every term
Multiply polynomials, including a binomial by a trinomial, using the distributive property or an area model
Explain what it means for polynomials to be closed under addition, subtraction and multiplication, and why they are not closed under division
Compare polynomial arithmetic with integer arithmetic and use substitution to check a result
Prior Knowledge Required
Students should already be comfortable with:
Applying the properties of operations to generate equivalent expressions 6.EE.A.3
Adding, subtracting, factoring and expanding linear expressions 7.EE.A.1
Using the product rule for exponents, xm · xn = xm+n8.EE.A.1
Adding, subtracting and multiplying integers, including negative integers
"Compute 312 + 145. Then add (3x2 + x + 2) + (x2 + 4x + 5). What happens to your polynomial answer when you replace x with 10?"
Students get 457 and 4x2 + 5x + 7, which equals 400 + 50 + 7 = 457 when x = 10. Ask what played the role of the hundreds, tens and ones columns (the x2, x and constant terms). Then ask two quick questions: "Is 7 - 12 an integer? Is 7 ÷ 12 an integer?" Tell students the lesson will ask the same questions about polynomials.
Direct Instruction20 minutes
Define a polynomial as a sum of terms of the form axn, where a is a number and n is a whole number (0, 1, 2, ...). Then model each operation:
Add: group like terms (same variable, same exponent) and add their coefficients. The exponents do not change.
Subtract: rewrite subtraction as adding the opposite, so the minus sign changes the sign of every term in the second polynomial, then add.
Multiply: multiply every term of the first polynomial by every term of the second (the distributive property), add exponents on x, then combine like terms.
Check: substitute a number such as x = 1 or x = 10 into the original expression and your answer. The results must match.
After the examples, state closure directly: each answer is again a sum of terms with whole-number exponents, so it is a polynomial. Then show the exception: x ÷ (x + 1) cannot be written as a polynomial, just as 3 ÷ 4 is not an integer (Diagram 2).
Guided Practice15 minutes
Pairs solve three problems on whiteboards and check each one by substituting x = 1. (1) (6x2 - x + 4) - (2x2 + 3x - 5) = 4x2 - 4x + 9. (2) (x + 5)(x - 5) = x2 - 25. (3) (x - 1)(x2 + x + 1) = x3 - 1. After each problem, ask a pair to explain why their answer is a polynomial. Watch for students who subtract only the first term of the second polynomial.
Independent Practice15 minutes
Students complete five problems alone: two additions or subtractions with three or more terms, two products (one binomial by binomial and one binomial by trinomial), and one "explain" prompt: "Give an example showing that the quotient of two polynomials is not always a polynomial." Students check each computation by substitution.
Closure5-10 minutes
Exit ticket: "(a) Find (x + 3)(x2 - 2x + 4). (b) Complete the sentence: The product of two polynomials is always a polynomial because ____." (Answer: x3 + x2 - 2x + 12; every product of terms is a number times x to a whole-number power, and adding such terms gives a polynomial.)
Differentiation Strategies
For Struggling Students
Use algebra tiles or a grid (area model) for every product until students can list all partial products without it
Have students rewrite each subtraction as adding the opposite, writing the changed signs in a second color
Color-code like terms (for example, circle x2 terms and underline x terms) before combining
For Advanced Students
Explain why the degree of a product of two nonzero polynomials is the sum of their degrees, but the degree of a sum can be smaller than either
Multiply (x2 + 3x + 2)(x2 - 3x + 2) and look for a shortcut using the structure of the factors
Explore 17 × 16 and (x + 7)(x + 6) at x = 10: why do polynomial coefficients not need carrying?
Assessment Guidance
What to Look For
Look for correct distribution of a subtraction sign across every term, correct exponents in products (x · x2 = x3), and no combining of unlike terms such as x2 and x. For the closure part of the standard, students should be able to explain in words why the result of +, - or × is a polynomial and give an example of division leaving the system; a correct computation alone does not show that understanding.
02
Classroom Activities
3 Activities
1
Integer Twins
20 minPairs
Each pair gets cards with an integer calculation and its polynomial "twin" (the same digits used as coefficients, with x = 10). One partner does the integer calculation, the other does the polynomial calculation, and they check that the polynomial answer at x = 10 equals the integer answer.
Card Examples
243 + 516 and (2x2 + 4x + 3) + (5x2 + x + 6) = 7x2 + 5x + 9, which is 759 at x = 10
587 - 234 and (5x2 + 8x + 7) - (2x2 + 3x + 4) = 3x2 + 5x + 3, which is 353 at x = 10
21 × 13 and (2x + 1)(x + 3) = 2x2 + 7x + 3, which is 273 at x = 10
17 × 16 and (x + 7)(x + 6) = x2 + 13x + 42, which is 272 at x = 10
Discussion Questions
In the last card the coefficients 13 and 42 are bigger than 9. What does the integer calculation do instead, and why does the polynomial not need it?
Does checking at x = 10 prove the polynomial answer is right? What would a second check value add?
Modification for Distance Learning
Share the cards on a slide and have pairs type both answers into a shared table, with a column for the value at x = 10.
2
Closure Court
20 minGroups of 3-4
Groups receive claim cards about polynomials. For each claim they decide "always true" or "not always true" and must support the verdict with an argument or a counterexample.
Claim Cards
"The product of two polynomials is a polynomial." (Always true)
"The sum of two polynomials of degree 3 has degree 3." (Not always: (x3 + x) + (-x3 + 2) = x + 2)
"The difference of two binomials is a binomial." (Not always: (x + 2) - (x - 3) = 5)
"The quotient of two polynomials is a polynomial." (Not always: 1 ÷ x is not a polynomial)
"The degree of a product of two nonzero polynomials is the sum of their degrees." (Always true)
Procedure
Groups have 12 minutes to rule on every card and write their evidence
Each group presents one ruling; other groups may object with a counterexample
Close by matching each polynomial claim with the corresponding claim about integers
3
Garden Path Design
20 minIndividual then share
Students write polynomial expressions for a garden plan and use addition, subtraction and multiplication to find areas and perimeters. They then choose a value of x and check every expression with actual measurements.
The Plan
A rectangular garden bed is 2x + 1 feet long and x + 3 feet wide, so its area is (2x + 1)(x + 3) = 2x2 + 7x + 3 square feet
A path 2 feet wide surrounds the bed, so the outer rectangle is 2x + 5 by x + 7, with area 2x2 + 19x + 35
The path alone has area (2x2 + 19x + 35) - (2x2 + 7x + 3) = 12x + 32 square feet
Share and Check
With x = 4, the bed is 9 ft by 7 ft (63 sq ft), the outer rectangle is 13 ft by 11 ft (143 sq ft), and the path is 80 sq ft, which matches 12(4) + 32
Students write the outer perimeter as a polynomial, 2(2x + 5) + 2(x + 7) = 6x + 24, and check it with x = 4
Gallery Variation
Students design their own garden with a path and post the polynomial for the path area. Classmates pick a value of x, measure the design on grid paper and confirm the polynomial.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Area Models for 23 × 14 and (2x + 3)(x + 4)
Each rectangle is split into partial products. The integer model and the polynomial model are drawn to the same scale with x = 10, so every piece matches: 2x2 is 200, 3x is 30, 8x is 80 and 12 is 12. Combining the like terms 3x and 8x is the same step as adding the tens.
Diagram 2: Closure for Integers and Polynomials
Integers and polynomials are both closed under addition, subtraction and multiplication, and neither is closed under division. Green marks closed operations and red marks the operation that can leave the system.
04
Homework Assignment
~30 min
HSA.APR.A.1 Homework: Polynomial Arithmetic
Directions: Simplify each expression and write the answer in standard form (highest power first). Show every step. Check each answer by substituting x = 1 (or y = 1) into the original expression and into your answer. For Problem 6(b), answer in complete sentences.
Part 1: Adding and Subtracting (Problems 1-3)
(5x2 + 3x - 8) + (2x2 - 7x + 4)
(6y3 - 2y + 9) - (4y3 + y2 - 5y - 1)
A triangle has sides of length 2x + 5, x2 - 3 and 3x2 + x - 1 centimeters. Write its perimeter as a polynomial in standard form.
Part 2: Multiplying (Problems 4-5)
(3x - 4)(2x + 5)
(x2 - 2x + 3)(x2 + x - 1)
Part 3: Closure in Context (Problem 6)
An open box is made from a 20-inch by 30-inch sheet of cardboard by cutting a square of side x inches from each corner and folding up the sides. (a) Write the volume V = x(20 - 2x)(30 - 2x) as a polynomial in standard form, and check it at x = 3. (b) Explain why your answer must be a polynomial, using the idea of closure. Then give an example of an operation on two polynomials whose result is not a polynomial.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Combining Like Terms
Only like terms combined, correct coefficients
One sign or coefficient error
Unlike terms combined or exponents changed
Distribution
Subtraction sign and every product distributed correctly
One term missed in distribution
Distribution not used or mostly incorrect
Check by Substitution
Check shown and values match
Check shown with an arithmetic error
No check
Closure Explanation
Clear reason and a correct non-example for division
Reason given without a non-example, or vice versa
No explanation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
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
Simplify (4x2 - 3x + 2) + (x2 + 5x - 9).
Answer: C
Combine like terms: 4x2 + x2 = 5x2, -3x + 5x = 2x, and 2 - 9 = -7. Choice B adds the exponents, but adding like terms never changes the exponent. Choice A treats -3x as +3x, and choice D computes 2 - 9 as 11.
Question 2 of 20 · Multiple Choice
Simplify (7x2 + 2x - 1) - (3x2 - 4x + 6).
Answer: A
The subtraction changes the sign of every term in the second polynomial: 7x2 + 2x - 1 - 3x2 + 4x - 6 = 4x2 + 6x - 7. Choice B subtracts only the first term and adds the rest. Choice C adds the two polynomials instead of subtracting.
Question 3 of 20 · Multiple Choice
Multiply (x + 6)(x - 2).
Answer: B
x · x = x2, x · (-2) = -2x, 6 · x = 6x and 6 · (-2) = -12, so the product is x2 + 4x - 12. Choice A multiplies only the first terms and the last terms. Choice D has the sign of the middle term reversed.
Question 4 of 20 · Multiple Choice
Which expression is equal to (2x - 5)2?
Answer: D
(2x - 5)2 = (2x - 5)(2x - 5) = 4x2 - 10x - 10x + 25 = 4x2 - 20x + 25. Choice A squares each term separately and loses the middle term. Choice C forgets to square the coefficient 2.
Question 5 of 20 · Multiple Choice
For which operation is the set of polynomials not closed?
Answer: D
Adding, subtracting or multiplying two polynomials always gives a polynomial. Dividing may not: x ÷ (x + 1) cannot be written as a polynomial. This matches the integers, where 3 ÷ 4 is not an integer. Choice C is a common guess because products can have high degree, but the result is still a polynomial.
Question 6 of 20 · Multiple Choice
Which of these is a polynomial?
Answer: C
A polynomial has only whole-number exponents on the variable. (2/3)x4 - 7x + 3 qualifies; a fractional coefficient such as 2/3 is allowed. Choice A has a negative exponent, choice B is (2x)1/2 - 1, and choice D is 4x-2 + x.
Question 7 of 20 · Multiple Choice
What is the degree of (3x4 + x)(2x3 - 7)?
Answer: B
The highest-degree term of the product is 3x4 · 2x3 = 6x7, so the degree is 4 + 3 = 7. Choice A multiplies the degrees instead of adding the exponents. Choice C is the degree of the first factor only.
Question 8 of 20 · Multiple Choice
Multiply (x - 3)(x2 + 3x + 9).
Answer: A
x(x2 + 3x + 9) - 3(x2 + 3x + 9) = x3 + 3x2 + 9x - 3x2 - 9x - 27 = x3 - 27. The middle terms cancel. Choice B has the wrong sign on the constant, since -3 · 9 = -27. Choice D adds the x2 terms instead of cancelling them.
Question 9 of 20 · Multiple Choice
Polynomials are closed under multiplication. Which statement about integers is the matching fact?
Answer: B
Closure under multiplication means the product of any two members is again a member, and the product of two integers is always an integer. Choice A is false (3 ÷ 4), choice C is false (3 + 5 = 8), and choice D is false (√2 is not an integer).
Question 10 of 20 · Multiple Choice
Simplify 5 - (x2 - 2x + 3).
Answer: C
Distribute the minus sign: 5 - x2 + 2x - 3 = -x2 + 2x + 2. Choice A changes the sign of x2 but not of -2x. Choice D adds 3 instead of subtracting it.
Question 11 of 20 · Multiple Choice
Multiply -3x(2x2 - x + 4).
Answer: A
-3x · 2x2 = -6x3, -3x · (-x) = 3x2 and -3x · 4 = -12x. Choice B misses that a negative times a negative is positive. Choice C forgets to add exponents, and choice D does not multiply the last term.
Question 12 of 20 · Multiple Choice
A rectangle is 3x + 2 units long and x - 1 units wide. Which polynomial gives its area?
Answer: D
Area = (3x + 2)(x - 1) = 3x2 - 3x + 2x - 2 = 3x2 - x - 2. Choice A multiplies only first and last terms. Choice B adds the two sides, which is half the perimeter. Choice C adds the middle terms as if both were positive.
Question 13 of 20 · Multiple Choice
What is the coefficient of x2 in the product (x2 + 2x - 1)(3x - 4)?
Answer: B
Two partial products give x2 terms: x2 · (-4) = -4x2 and 2x · 3x = 6x2. Their sum is 2x2, so the coefficient is 2. The full product is 3x3 + 2x2 - 11x + 4. Choices A and C each use only one of the two partial products, and choice D is the coefficient of x3.
Question 14 of 20 · Multiple Choice
A student writes (x + 4)2 = x2 + 16. Which response is correct?
Answer: C
(x + 4)(x + 4) = x2 + 4x + 4x + 16 = x2 + 8x + 16. Substituting x = 1 shows the student's version is wrong: 52 = 25, but 1 + 16 = 17. Choice B counts the middle product only once. Choice D doubles instead of squaring.
Question 15 of 20 · Short Answer
Simplify (2a2 - 3a) + (a2 + 4) - (5a2 - a - 1).
Change the signs in the last polynomial: 2a2 - 3a + a2 + 4 - 5a2 + a + 1. Combine like terms: -2a2 - 2a + 5. Check at a = 1: the original is (2 - 3) + (1 + 4) - (5 - 1 - 1) = -1 + 5 - 3 = 1, and the answer is -2 - 2 + 5 = 1.
Question 16 of 20 · Short Answer
Multiply (2x + 1)(x2 - 3x + 4) and write the answer in standard form.
Multiply (3x + 2)(x + 4). Then use your answer with x = 10 to find 32 × 14, and explain why this works.
(3x + 2)(x + 4) = 3x2 + 14x + 8. At x = 10: 300 + 140 + 8 = 448, and 32 × 14 = 448. It works because 32 = 3(10) + 2 and 14 = 10 + 4, so the integer product is the polynomial product evaluated at x = 10. Base-ten numbers are polynomials in 10. The coefficient 14 is larger than 9, so the integer calculation carries 1 into the hundreds place, but the polynomial answer does not need to.
Question 18 of 20 · Short Answer
Explain why the sum of any two polynomials is always a polynomial. Use an example.
Each polynomial is a sum of terms axn with whole-number exponents. Adding two polynomials only groups terms with the same exponent and adds their coefficients, which are numbers, so every term of the result still has the form axn with a whole-number exponent. Example: (x3 + 2x) + (4x2 - 2x + 1) = x3 + 4x2 + 1, a polynomial. So polynomials are closed under addition, just as integers are.
A club sells x custom mugs at a price of 20 - 0.5x dollars each (for up to 30 mugs). Its costs are 4x + 50 dollars. Write the profit as a polynomial, and find the profit when 10 mugs are sold.
A set is closed under an operation if using the operation on any two members of the set always gives another member of the set. The integers are closed under addition, subtraction and multiplication, but not division, since 3 ÷ 4 is not an integer. HSA.APR.A.1 asks students to understand that polynomials have exactly the same closure properties.
Why does the standard compare polynomials to integers?
Because the arithmetic really is parallel. An integer in base ten is a polynomial in 10: 347 = 3(10)2 + 4(10) + 7. Adding like terms matches adding place values, and multiplying polynomials matches long multiplication without carrying. The comparison gives students a way to check work (substitute x = 10) and prepares them for later topics, such as polynomial division, that also mirror integer arithmetic.
What counts as a polynomial?
A sum of terms of the form axn, where the coefficient a is a number and the exponent n is a whole number (0, 1, 2, ...). So 5x3 - x + 1/2 is a polynomial, but 3x-2, √x and 2/x are not, because their exponents are negative or fractional. Polynomials can also have more than one variable, such as x2y + 3y.
Why aren't polynomials closed under division?
Dividing one polynomial by another can produce an expression with the variable in a denominator. For example, x ÷ (x + 1) is not equal to any polynomial. Some quotients do work out, such as (x2 - 1) ÷ (x - 1) = x + 1, but closure requires the result to be a polynomial every time. Rational expressions, which include such quotients, are studied in HSA.APR.D.6 and HSA.APR.D.7.
What are the common mistakes in polynomial arithmetic?
Subtracting only the first term of the second polynomial instead of every term
Adding exponents when adding terms (x2 + x2 = 2x2, not x4)
Combining unlike terms such as 3x2 and 2x
Writing (a + b)2 as a2 + b2, which loses the middle term 2ab
Missing a partial product when multiplying a binomial by a trinomial
Should students use FOIL?
FOIL (first, outer, inner, last) only works for a binomial times a binomial. The distributive property, or an area model, works for any product and makes the count of partial products clear: a binomial times a trinomial has 2 × 3 = 6 partial products. If students already know FOIL, connect it to the distributive property rather than banning it.
Do students have to prove closure formally?
No formal proof is expected, but students should be able to explain it in words: adding or multiplying terms of the form axn gives terms of the same form, because coefficients combine as numbers and whole-number exponents add to whole numbers. Asking for a non-example under division is a good test of understanding.
How can students check their answers?
Substitute a number into the original expression and into the answer. If the values differ, there is an error. x = 1 is quick; x = 10 connects to integer arithmetic; x = 2 catches errors that x = 1 can hide, such as confusing x2 and x3. Graphing both expressions on the same screen is another check.
How is HSA.APR.A.1 tested?
Typical items ask students to add, subtract or multiply polynomials and choose the equivalent expression, to find a coefficient or the degree of a product, or to write an expression for an area or a profit. The digital SAT's Advanced Math domain includes questions on equivalent polynomial expressions of this kind. Closure is usually tested with an "always, sometimes or never" item or a request for an example.
What comes after HSA.APR.A.1?
HSA.APR.B.2: the Remainder Theorem for division by x - a
HSA.APR.C.4: proving polynomial identities such as (x2 + y2)2 = (x2 - y2)2 + (2xy)2
HSA.APR.D.6: rewriting rational expressions, where division is handled
HSA.SSE.B.3: choosing an equivalent form, such as a factored form, to reveal properties of an expression
07
Related Standards
6 standards
These standards connect to HSA.APR.A.1: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
6.EE.A.3Prerequisite
Apply the properties of operations to generate equivalent expressions
