1 topic 6.1.3 subtracting polynomials. 2 lesson 1.1.1 california standards: 2.0 students understand...

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Topic 6.1.3Topic 6.1.3

Subtracting PolynomialsSubtracting Polynomials

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Lesson

1.1.1

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.

Subtracting PolynomialsSubtracting PolynomialsTopic

6.1.3

Key words:• polynomial• like terms• inverse

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Lesson

1.1.1

Subtracting one polynomial from another follows the same rules as adding polynomials.

Subtracting PolynomialsSubtracting PolynomialsTopic

6.1.3

You just need to combine like terms, then carry out all the subtractions to simplify the expression.

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Lesson

1.1.1

Subtracting Polynomials

Subtracting PolynomialsSubtracting PolynomialsTopic

6.1.3

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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Polynomial B – Polynomial A = x2 + 4x – (x2) – (x)

= x2 – x2 + 4x – x

= 0 + 3x

= 3x

Subtracting PolynomialsSubtracting Polynomials

Example 1

Topic

6.1.3

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:

Solution follows…

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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.

Lesson

1.1.1

Guided Practice

Subtracting PolynomialsSubtracting PolynomialsTopic

6.1.3

Solution follows…

x2 – x2 + 8 – (–4) = 12

8x2 – 5x – 3x + 4 – (–4)= 8x2 – 8x + 8

x2 – x – x – 4 = x2 – 2x – 4

–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

x2 – x2 + 8 – (–16) = 24

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Simplify:

8. (9a – 10) – (5a + 2)

9. (5a2 – 2a + 3) – (3a + 5)

10. (x3 + 5x2 – x) – (x2 + x)

Lesson

1.1.1

Guided Practice

Subtracting PolynomialsSubtracting PolynomialsTopic

6.1.3

Solution follows…

9a – 5a – 10 – 2 = 4a – 12

5a2 – 2a – 3a + 3 – 5 = 5a2 – 5a – 2

x3 + 5x2 – x2 – x – x = x3 + 4x2 – 2x

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Lesson

1.1.1

Subtracting is Simply Adding the Opposite

Subtracting PolynomialsSubtracting PolynomialsTopic

6.1.3

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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Subtracting PolynomialsSubtracting Polynomials

Example 2

Topic

6.1.3

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

Solution follows…

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Lesson

1.1.1

Subtracting is Simply Adding the Opposite

Subtracting PolynomialsSubtracting PolynomialsTopic

6.1.3

Alternatively, you can do subtraction by lining up terms vertically — this is shown in Example 3.

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–7x2 + 3x + 15)

+ (5x2 – 3x + 18)

–2x2 – 2x + 13)

–7x2 + 3x + 15 

– (–5x2 + 3x – 18)

–2x2 – 2x + 13)

Subtracting PolynomialsSubtracting Polynomials

Example 3

Topic

6.1.3

Subtract –5x2 + 3x – 8 from –7x2 + x + 5.

Solution

Solution follows…

OR

This is the opposite of –5x2 + 3x – 8

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Lesson

1.1.1

Guided Practice

Subtracting PolynomialsSubtracting PolynomialsTopic

6.1.3

Solution follows…

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)

15. 6x2 – 6 –(5x2 + 9)

16. 8a2 + 4a – 9 –(3a2 – 3a + 7)

8a4 – 2a2 + 4

–9x4 – x3 + 6x2 – 3x + 12

–4c4 + 17c2 + 13c + 5

–a2 + a – 4

x2 – 15 5a2 + 7a – 16

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17. Subtract 7a3 + 3a – 12 from 5a2 – a + 4 by adding the opposite expression. Use the vertical lining up method.

Lesson

1.1.1

Guided Practice

Subtracting PolynomialsSubtracting PolynomialsTopic

6.1.3

Solution follows…

18. Subtract (8p3 – 11p2 – 3p) from 4p3 + 6p2 – 10 by adding the opposite expression. Use the vertical lining up method.

–7a3 + 5a2 – 4a + 16

5a2 – 3a + 14+ –7a3 – 5a2 – 3a + 12

4p3 + 16p2 + 3p – 10

+ –8p3 + 11p2 + 3p – 10

–4p3 + 17p2 + 3p – 10

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Lesson

1.1.1

Subtracting is Simply Adding the Opposite

Subtracting PolynomialsSubtracting PolynomialsTopic

6.1.3

You know that when you add a number to its opposite, the result will always be 0.

It’s the same with polynomials — if you add a polynomial to its opposite, the result will always be 0.

3 + (–3) = 0

3x2 + 2x + 1+ –3x2 – 2x – 1

0

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–5x2 + 3x – 1 + (5x2 – 3x + 1)

= –5x2 + 3x – 1 + 5x2 – 3x + 1

= –5x2 + 5x2 + 3x – 3x – 1 + 1

= 0 + 0 + 0

= 0

Subtracting PolynomialsSubtracting Polynomials

Example 4

Topic

6.1.3

Find the sum of –5x2 + 3x – 1 and 5x2 – 3x + 1.

Solution

Solution follows…

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Subtracting PolynomialsSubtracting Polynomials

Independent Practice

Solution follows…

Topic

6.1.3

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)]

7.7. –10a2 + 4a – 1) – (7a2 + 4a) – 1

2a + 6

–6y

–6x2 + 11x – 9

a2 + a + 3

–x4 – x3 – 3x2 + 9x – 4 k + 3

–17a2 – 1 –x2 + 2x + 2

8. (x2 + 4x + 6) – (2x2 + 2x + 4)

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Subtracting PolynomialsSubtracting Polynomials

Independent Practice

Solution follows…

Topic

6.1.3

Solve these by first simplifying the left side of the equations.

9. (2x + 3) – (x – 7) = 40 10. (4x + 14) – (–10x – 3) = 73

11. (2 – 3x) – (7 – 2x) = 23

12. (17 – 5x) – (4 – 3x) – (6 – x) = 19

Find the opposite of the polynomials below.

13. x2 + 2x + 1 14. –a2 + 6a + 4

15. 4b2 – 6bc + 7c 16. a3 + 4a2 + 3a – 2

x = 30

x = – 28

x = 4

x = – 12

–x2 – 2x – 1 a2 – 6a – 4

–4b2 + 6bc – 7c –a3 – 4a2 – 3a + 2

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Subtracting PolynomialsSubtracting Polynomials

Independent Practice

Solution follows…

Topic

6.1.3

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

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Topic

6.1.3

Round UpRound Up

Subtracting PolynomialsSubtracting Polynomials

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.