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Algebra I

Adding and Subtracting Polynomials

Lesson

A polynomial is a sum of terms like 3x23x^2, −2x-2x, or 55. Adding or subtracting polynomials is just combining like terms — the same skill as before, but now we have terms with different exponents.

Two terms are like terms only if they have the same variable raised to the same exponent:

  • 3x23x^2 and 5x25x^2 are like (both x2x^2).
  • 3x23x^2 and 5x5x are NOT like (different exponents).
  • Constants are like terms with each other.

To subtract a polynomial, distribute the negative sign across every term inside the parentheses, then combine like terms.

Worked example 1 — adding

(3x2+2x−5)+(x2−4x+7)(3x^2 + 2x - 5) + (x^2 - 4x + 7)

Group like terms:

=(3x2+x2)+(2x−4x)+(−5+7)= (3x^2 + x^2) + (2x - 4x) + (-5 + 7)

Combine each group:

=4x2−2x+2= 4x^2 - 2x + 2

Worked example 2 — subtracting

(5x2−3x+1)−(2x2+x−4)(5x^2 - 3x + 1) - (2x^2 + x - 4)

Distribute the negative sign across the second polynomial — every sign flips:

=5x2−3x+1−2x2−x+4= 5x^2 - 3x + 1 - 2x^2 - x + 4

Combine like terms:

=3x2−4x+5= 3x^2 - 4x + 5

How to type your answer

Write terms in descending order of exponent (highest first). Use ^ for exponents. No spaces. Use x for 1x1x. Examples: 4x^2-2x+2, 3x^2-4x+5, x^2+5.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

(x+2)+(x+3)(x + 2) + (x + 3)

Problem 2

(2x)+(3x)(2x) + (3x)

Problem 3

(x2+1)+(x2+4)(x^2 + 1) + (x^2 + 4)

Problem 4

(3x+5)+(−x+2)(3x + 5) + (-x + 2)

Practice

Standard problems matching the lesson.

Problem 5

(3x2+2x)+(x2+5x)(3x^2 + 2x) + (x^2 + 5x)

Problem 6

(2x2−3)+(4x2+7)(2x^2 - 3) + (4x^2 + 7)

Problem 7

(x2+3x−2)+(2x2−x+5)(x^2 + 3x - 2) + (2x^2 - x + 5)

Problem 8

(5x+4)+(−2x+9)(5x + 4) + (-2x + 9)

Problem 9

(3x2+2x−5)+(x2−4x+7)(3x^2 + 2x - 5) + (x^2 - 4x + 7)

Problem 10

(7x2−4x+3)+(−3x2+6x−5)(7x^2 - 4x + 3) + (-3x^2 + 6x - 5)

Problem 11

(5x2−3x)−(2x2+x)(5x^2 - 3x) - (2x^2 + x)

Problem 12

(4x+7)−(x−3)(4x + 7) - (x - 3)

Problem 13

(5x2−3x+1)−(2x2+x−4)(5x^2 - 3x + 1) - (2x^2 + x - 4)

Problem 14

(8x2+2x−6)−(3x2−5x+4)(8x^2 + 2x - 6) - (3x^2 - 5x + 4)

Problem 15

(x2−9)−(x2+4)(x^2 - 9) - (x^2 + 4)

Problem 16

(2x3+x2)+(x3−4x2)(2x^3 + x^2) + (x^3 - 4x^2)

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

(4x3+2x−1)+(−x3+3x+5)(4x^3 + 2x - 1) + (-x^3 + 3x + 5)

Problem 18

(5x2−2x+8)−(−x2+4x−3)(5x^2 - 2x + 8) - (-x^2 + 4x - 3)

Problem 19

(7x2+3)+(2x−5)−(4x+1)(7x^2 + 3) + (2x - 5) - (4x + 1)

Problem 20

(2x3−x2+4)+(x3+x2−6)(2x^3 - x^2 + 4) + (x^3 + x^2 - 6)

Problem 21

(3x2+2x−1)−(3x2−4x+5)(3x^2 + 2x - 1) - (3x^2 - 4x + 5)

Problem 22

(6x2−5)−(2x2+3x)+(x2+3x)(6x^2 - 5) - (2x^2 + 3x) + (x^2 + 3x)

Practice

Standard problems matching the lesson.

Problem 23

Lot A area is 4x^2 + x sq ft; Lot B is 2x^2 + 3x sq ft. Find the combined area as a polynomial.

Problem 24

A rectangle has length (2x + 5) and width (x + 3). Simplify the perimeter 2(2x + 5) + 2(x + 3).

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

A region has area 5x^2 + 3x - 1 sq m; a sub-region of 2x^2 - x + 4 is removed. Find the remaining area.

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