Transcript Subtracting Polynomials Topic 6.1.3 Lesson Topic 1.1.1 6.1.3 Subtracting Polynomials California Standards: What it means for you: 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking.

Subtracting Polynomials
Topic 6.1.3
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Lesson
Topic
1.1.1
6.1.3
Subtracting Polynomials
California Standards:
What it means for you:
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.
You’ll learn how to subtract
polynomials.
Key words:
• polynomial
• like terms
• inverse
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Lesson
Topic
1.1.1
6.1.3
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.
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Lesson
Topic
1.1.1
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.
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Topic
6.1.3
Example
Subtracting Polynomials
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
=
0
+
3x
= 3x
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Solution follows…
Lesson
Topic
1.1.1
6.1.3
Subtracting Polynomials
Guided Practice
1. Subtract x2 – 4 from x2 + 8.
x2 – x2 + 8 – (–4) = 12
2. Subtract 3x – 4 from 8x2 – 5x + 4.
8x2 – 5x – 3x + 4 – (–4)
= 8x2 – 8x + 8
3. Subtract x + 4 from x2 – x.
x2 – x – x – 4 = x2 – 2x – 4
4. Subtract x2 – 16 from x2 + 8.
x2 – x2 + 8 – (–16) = 24
5. Subtract x2 + x – 1 from x + 4.
–x2 + x – x + 4 – (–1) = –x2 + 5
6. Subtract –3x2 + 4x – 5 from x2 – 7. x2 – 2(–3x2) – 4x – 7 – (–5)
= 4x – 4x – 2
7. Subtract –3x2 – 5x + 2 from –2x3 – x2 – 7x.
–2x3 – x2 – (–3x2) – 7x – (–5x) – 2
= –2x3 + 2x2 – 2x – 2
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Solution follows…
Lesson
Topic
1.1.1
6.1.3
Subtracting Polynomials
Guided Practice
Simplify:
8. (9a – 10) – (5a + 2)
9a – 5a – 10 – 2 = 4a – 12
9. (5a2 – 2a + 3) – (3a + 5)
5a2 – 2a – 3a + 3 – 5 = 5a2 – 5a – 2
10. (x3 + 5x2 – x) – (x2 + x)
x3 + 5x2 – x2 – x – x = x3 + 4x2 – 2x
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Solution follows…
Lesson
Topic
1.1.1
6.1.3
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.
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Topic
6.1.3
Example
Subtracting Polynomials
2
Subtract –5x2 + 3x – 8 from –7x2 + x + 5.
Solution
–7x2 + x + 5 – (–5x2 + 3x – 8)
= –7x2 + x + 5 + 5x2 – 3x + 8
= –7x2 + 5x2 + x – 3x + 5 + 8
= –2x2 – 2x + 13
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Solution follows…
Lesson
Topic
1.1.1
6.1.3
Subtracting Polynomials
Subtracting is Simply Adding the Opposite
Alternatively, you can do subtraction by lining up terms
vertically — this is shown in Example 3.
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Topic
6.1.3
Example
Subtracting Polynomials
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)
OR
+ (5x2 – 3x + 18)
–2x2 – 2x + 13)
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Solution follows…
Lesson
Topic
1.1.1
6.1.3
Subtracting Polynomials
Guided Practice
Simplify the expressions in Exercises 11–16.
11. (3a4 + 4) – (2a2 – 5a4)
8a4 – 2a2 + 4
12. (6x2 + 8 – 9x4) – (3x – 4 + x3)
–9x4 – x3 + 6x2 – 3x + 12
13. (9c2 + 11c2 + 5c – 5) – (–10 + 4c4 – 8c + 3c2)
–4c4 + 17c2 + 13c + 5
14. (8a2 – 2a + 5a) – (9a2 + 2a + 4)
15. 6x2 – 6
–(5x2 + 9)
x2 – 15
16.
–a2 + a – 4
8a2 + 4a – 9
–(3a2 – 3a + 7)
5a2 + 7a – 16
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Solution follows…
Lesson
Topic
1.1.1
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.
5a2 – 3a + 14
+ –7a3 – 5a2 – 3a + 12
–7a3 + 5a2 – 4a + 16
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
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Solution follows…
Lesson
Topic
1.1.1
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.
3x2 + 2x + 1
+ –3x2 – 2x – 1
0
3 + (–3) = 0
It’s the same with polynomials
— if you add a polynomial to its
opposite, the result will always be 0.
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Topic
Subtracting Polynomials
6.1.3
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 – 1 + 1
=
0
+
0
+
0
=0
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Solution follows…
Topic
Subtracting Polynomials
6.1.3
Independent Practice
Subtract the polynomials and simplify the resulting expression.
1. (5a + 8) – (3a + 2)
2a + 6
2. (8x – 2y) – (8x + 4y)
–6y
3. (–4x2 + 7x – 3) – (2x2 – 4x + 6)
4. (3a2 + 2a + 6) – (2a2 + a + 3)
–6x2 + 11x – 9
a2 + a + 3
5. –3x4 – 2x3 + 4x – 1 – (–2x4 – x3 + 3x2 – 5x + 3)
6. 5 – [(2k + 3) – (3k + 1)]
7. –10a2 + 4a – 1)
– (7a2 + 4a) – 1
–17a2
–1
–x4 – x3 – 3x2 + 9x – 4
k+3
8. (x2 + 4x + 6)
– (2x2 + 2x + 4)
–x2 + 2x + 2
16
Solution follows…
Topic
6.1.3
Subtracting Polynomials
Independent Practice
Solve these by first simplifying the left side of the equations.
9. (2x + 3) – (x – 7) = 40
x = 30
10. (4x + 14) – (–10x – 3) = 73
x=4
11. (2 – 3x) – (7 – 2x) = 23
x = – 28
12. (17 – 5x) – (4 – 3x) – (6 – x) = 19
x = – 12
Find the opposite of the polynomials below.
13. x2 + 2x + 1
–x2 – 2x – 1
15. 4b2 – 6bc + 7c
–4b2 + 6bc – 7c
14. –a2 + 6a + 4
a2 – 6a – 4
16. a3 + 4a2 + 3a – 2
–a3 – 4a2 – 3a + 2
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Solution follows…
Topic
6.1.3
Subtracting Polynomials
Independent Practice
17. The opposite of a fifth degree polynomial has
5th degree
what degree?
18. If a monomial is subtracted from another monomial,
what are the possible results? A binomial, if the terms are not like
terms, or another monomial if the
terms are like terms.
19. What is the degree of the polynomial formed when
a 2nd degree polynomial is subtracted from a 1st
2nd degree
degree polynomial?
20. A 3rd degree polynomial has a 2nd degree polynomial
subtracted from it. What is the degree of the resulting
3rd degree
polynomial?
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Solution follows…
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.
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