1. Subtracting Polynomials
Topic 6.1.3

2. Subtracting Polynomials
Lesson
1.1.1
Topic
6.1.3
Subtracting Polynomials
California Standards:
2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.
10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques.
What it means for you:
You’ll learn how to subtract polynomials.
Key words:
polynomial
like terms
inverse

3. Subtracting Polynomials
Topic
6.1.3
Lesson
1.1.1
Subtracting Polynomials
Subtracting one polynomial from another follows the same rules as adding polynomials.
You just need to combine like terms, then carry out all the subtractions to simplify the expression.

4. Subtracting Polynomials
Lesson
1.1.1
Topic
6.1.3
Subtracting Polynomials
Subtracting Polynomials
Subtracting polynomials is the same as subtracting numbers.
To subtract Polynomial A from Polynomial B, you need to subtract each term of Polynomial A from Polynomial B.
Then you can combine any like terms to simplify the expression.

5. Subtracting Polynomials
Topic
6.1.3
Subtracting Polynomials
Example 1
Subtract Polynomial A from Polynomial B, where Polynomial A = x2 + x and Polynomial B = x2 + 4x.
Solution
Subtract each term of Polynomial A from Polynomial B:
Polynomial B – Polynomial A = x2 + 4x – (x2) – (x) = x2 – x2 + 4x – x = x = 3x
Solution follows…

6. Subtracting Polynomials
Topic
6.1.3
Lesson
1.1.1
Subtracting Polynomials
Guided Practice
1. Subtract x2 – 4 from x2 + 8.
2. Subtract 3x – 4 from 8x2 – 5x + 4.
3. Subtract x + 4 from x2 – x.
4. Subtract x2 – 16 from x2 + 8.
5. Subtract x2 + x – 1 from x + 4.
6. Subtract –3x2 + 4x – 5 from x2 – 7.
7. Subtract –3x2 – 5x + 2 from –2x3 – x2 – 7x.
x2 – x2 + 8 – (–4) = 12
8x2 – 5x – 3x + 4 – (–4) = 8x2 – 8x + 8
x2 – x – x – 4 = x2 – 2x – 4
x2 – x2 + 8 – (–16) = 24
–x2 + x – x + 4 – (–1) = –x2 + 5
x2 – (–3x2) – 4x – 7 – (–5)
= 4x2 – 4x – 2
–2x3 – x2 – (–3x2) – 7x – (–5x) – 2
= –2x3 + 2x2 – 2x – 2
Solution follows…

7. Subtracting Polynomials
Lesson
1.1.1
Topic
6.1.3
Subtracting Polynomials
Guided Practice
Simplify:
8. (9a – 10) – (5a + 2)
9. (5a2 – 2a + 3) – (3a + 5)
10. (x3 + 5x2 – x) – (x2 + x)
9a – 5a – 10 – 2 = 4a – 12
5a2 – 2a – 3a + 3 – 5 = 5a2 – 5a – 2
x3 + 5x2 – x2 – x – x = x3 + 4x2 – 2x
Solution follows…

8. Subtracting Polynomials
Topic
6.1.3
Lesson
1.1.1
Subtracting Polynomials
Subtracting is Simply Adding the Opposite
Another way to look at subtraction of polynomials is to go back to the definition of subtraction.
When you subtract Polynomial A from Polynomial B, what you’re actually doing is adding the opposite of Polynomial A to Polynomial B.

9. Subtracting Polynomials
Topic
6.1.3
Subtracting Polynomials
Example 2
Subtract –5x2 + 3x – 8 from –7x2 + x + 5.
Solution
–7x2 + x + 5 – (–5x2 + 3x – 8) = –7x2 + x x2 – 3x + 8 = –7x2 + 5x2 + x – 3x = –2x2 – 2x + 13
Solution follows…

10. Subtracting Polynomials
Topic
6.1.3
Lesson
1.1.1
Subtracting Polynomials
Subtracting is Simply Adding the Opposite
Alternatively, you can do subtraction by lining up terms vertically — this is shown in Example 3.

11. Subtracting Polynomials
Topic
6.1.3
Subtracting Polynomials
Example 3
Subtract –5x2 + 3x – 8 from –7x2 + x + 5.
Solution
This is the opposite of –5x2 + 3x – 8
–7x2 + 3x + 15  – (–5x2 + 3x – 18) –2x2 – 2x + 13)
–7x2 + 3x + 15) + (5x2 – 3x + 18) –2x2 – 2x + 13) OR
Solution follows…

12. Subtracting Polynomials
Lesson
1.1.1
Topic
6.1.3
Subtracting Polynomials
Guided Practice
Simplify the expressions in Exercises 11–16.
11. (3a4 + 4) – (2a2 – 5a4)
12. (6x2 + 8 – 9x4) – (3x – 4 + x3)
13. (9c2 + 11c2 + 5c – 5) – (–10 + 4c4 – 8c + 3c2)
14. (8a2 – 2a + 5a) – (9a2 + 2a + 4)
8a4 – 2a2 + 4
–9x4 – x3 + 6x2 – 3x + 12
–4c4 + 17c2 + 13c + 5
–a2 + a – 4
x2 – –(5x2 + 9)
a2 + 4a – –(3a2 – 3a + 7)
x2 – 15
5a a – 16
Solution follows…

13. Subtracting Polynomials
Lesson
1.1.1
Topic
6.1.3
Subtracting Polynomials
Guided Practice
17. Subtract 7a3 + 3a – 12 from 5a2 – a + 4 by adding the opposite expression. Use the vertical lining up method.
–7a3 + 5a2 – 4a + 16
5a2 – 3a + 14
+ –7a3 – 5a2 – 3a + 12
18. Subtract (8p3 – 11p2 – 3p) from 4p3 + 6p2 – 10 by adding the opposite expression. Use the vertical lining up method.
4p3 + 16p2 + 3p – 10
+ –8p3 + 11p2 + 3p – 10
–4p3 + 17p2 + 3p – 10
Solution follows…

14. Subtracting Polynomials
Lesson
1.1.1
Topic
6.1.3
Subtracting Polynomials
Subtracting is Simply Adding the Opposite
You know that when you add a number to its opposite, the result will always be 0.
3 + (–3) = 0
3x2 + 2x + 1
+ –3x2 – 2x – 1
It’s the same with polynomials — if you add a polynomial to its opposite, the result will always be 0.

15. Subtracting Polynomials
Topic
6.1.3
Subtracting Polynomials
Example 4
Find the sum of –5x2 + 3x – 1 and 5x2 – 3x + 1.
Solution
–5x2 + 3x – 1 + (5x2 – 3x + 1) = –5x2 + 3x – 1 + 5x2 – 3x + 1 = –5x2 + 5x2 + 3x – 3x – = = 0
Solution follows…

16. Subtracting Polynomials
Topic
6.1.3
Subtracting Polynomials
Independent Practice
Subtract the polynomials and simplify the resulting expression.
1. (5a + 8) – (3a + 2)
2. (8x – 2y) – (8x + 4y)
3. (–4x2 + 7x – 3) – (2x2 – 4x + 6)
4. (3a2 + 2a + 6) – (2a2 + a + 3)
5. –3x4 – 2x3 + 4x – 1 – (–2x4 – x3 + 3x2 – 5x + 3)
6. 5 – [(2k + 3) – (3k + 1)] 7.
2a + 6
–6y
–6x2 + 11x – 9
a2 + a + 3
–x4 – x3 – 3x2 + 9x – 4
k + 3
7. –10a2 + 4a – 1)
– (7a2 + 4a) – 1
(x2 + 4x + 6) – (2x2 + 2x + 4)
–17a – 1
–x x + 2
Solution follows…

17. Subtracting Polynomials
Topic
6.1.3
Subtracting Polynomials
Independent Practice
Solve these by first simplifying the left side of the equations.
9. (2x + 3) – (x – 7) = (4x + 14) – (–10x – 3) = 73
11. (2 – 3x) – (7 – 2x) = 23
12. (17 – 5x) – (4 – 3x) – (6 – x) = 19
x = 30
x = 4
x = – 28
x = – 12
Find the opposite of the polynomials below.
13. x2 + 2x –a2 + 6a + 4
15. 4b2 – 6bc + 7c a3 + 4a2 + 3a – 2
–x2 – 2x – 1
a2 – 6a – 4
–4b2 + 6bc – 7c
–a3 – 4a2 – 3a + 2
Solution follows…

18. Subtracting Polynomials
Topic
6.1.3
Subtracting Polynomials
Independent Practice
17. The opposite of a fifth degree polynomial has what degree?
18. If a monomial is subtracted from another monomial, what are the possible results?
19. What is the degree of the polynomial formed when a 2nd degree polynomial is subtracted from a 1st degree polynomial?
20. A 3rd degree polynomial has a 2nd degree polynomial subtracted from it. What is the degree of the resulting polynomial?
5th degree
A binomial, if the terms are not like terms, or another monomial if the terms are like terms.
2nd degree
3rd degree
Solution follows…

19. Subtracting Polynomials
Topic
6.1.3
Subtracting Polynomials
Round Up
Watch out for the signs when you’re subtracting polynomials.
It’s usually a good idea to put parentheses around the polynomial you’re subtracting, to make it easier to keep track of the signs.