1 topic 6.1.2 adding polynomials. 2 lesson 1.1.1 california standards: 2.0 students understand and...
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Topic 6.1.2Topic 6.1.2
Adding PolynomialsAdding Polynomials
2
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 add polynomials and multiply a polynomial by a number.
Adding PolynomialsAdding PolynomialsTopic
6.1.2
Key words:• polynomial• like terms• inverse
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Lesson
1.1.1
You saw in Topic 6.1.1 that polynomials are just algebraic expressions with one or more terms.
Adding PolynomialsAdding PolynomialsTopic
6.1.2
Adding polynomials isn’t difficult at all.
The only problem is that you can only add certain parts of each polynomial together.
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Lesson
1.1.1
The Opposite of a Polynomial
Adding PolynomialsAdding PolynomialsTopic
6.1.2
The opposite of a number is its additive inverse.
The opposite of a positive number is its corresponding negative number, and vice versa.
For example, –1 is the opposite of 1, and 1 is the opposite of –1.
To find the opposite of a polynomial, you make the positive terms negative and the negative terms positive.
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a) –2x + 1 b) 5x2 – 3x + 1
Adding PolynomialsAdding Polynomials
Example 1
Topic
6.1.2
Find the opposites of the following polynomials:
a) 2x – 1 b) –5x2 + 3x – 1
Solution
Solution follows…
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Find the opposites of the following polynomials.
Lesson
1.1.1
Guided Practice
Adding PolynomialsAdding PolynomialsTopic
6.1.2
Solution follows…
1. 2x + 1 2. –5x – 1
3. x2 + 5x – 2 4. 3x2 – 2x + 3
5. 3x2 + 4x – 8 6. –8x2 – 4x + 4
7. 4x4 – 16 8. 8x3 – 6x2 + 6x – 8
9. 5x4 – 6x2 + 7 10. –2x4 + 3x3 – 2x2
11. –0.9x3 – 0.8x2 – 0.4x – 1.0 12. –1.4x3 – 0.8x2 – x12
–2x – 1
–x2 – 5x + 2
–3x2 – 4x + 8
–4x4 + 16
–5x4 + 6x2 – 7
0.9x3 + 0.8x2 + 0.4x + 1.0
5x + 1
–3x2 + 2x – 3
8x2 + 4x – 4
–8x3 + 6x2 – 6x + 8
2x4 – 3x3 + 2x2
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1.4x3 + 0.8x2 + x
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Lesson
1.1.1
Adding Polynomials
Adding PolynomialsAdding PolynomialsTopic
6.1.2
Adding polynomials consists of combining all like terms.
There are a few ways of adding polynomials — one method is by collecting like terms and simplifying, another is through the vertical lining up of terms.
The following Example explains these two methods.
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(–5x2 + 3x – 1) + (6x2 – x + 3) + (5x – 7)
= –5x2 + 3x – 1 + 6x2 – x + 3 + 5x – 7
= –5x2 + 6x2 + 3x – x + 5x – 1 + 3 – 7
= x2 + 7x – 5
Adding PolynomialsAdding Polynomials
Example 2
Topic
6.1.2
Find the sum of –5x2 + 3x – 1, 6x2 – x + 3, and 5x – 7.
Solution
Method A — Collecting Like Terms and Simplifying
Solution follows…
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–5x2 + 3x – 1
Adding PolynomialsAdding Polynomials
Example 2
Topic
6.1.2
Find the sum of –5x2 + 3x – 1, 6x2 – x + 3, and 5x – 7.
Solution
Method B — Vertical Lining Up of Terms
Solution continues…
Both methods give the same solution.
x2 + 7x – 5
+ 6x2 – x + 3
+ 5x – 7
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Lesson
1.1.1
Multiplying a Polynomial by a Number
Adding PolynomialsAdding PolynomialsTopic
6.1.2
Multiplying a polynomial by a number is the same as adding the polynomial together several times.
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Adding PolynomialsAdding Polynomials
Example 3
Topic
6.1.2
Multiply x + 3 by 3.
Solution
Solution follows…
(x + 3) × 3 = (x + 3) + (x + 3) + (x + 3)
= x + x + x + 3 + 3 + 3
= 3x + 9
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Lesson
1.1.1
Multiplying a Polynomial by a Number
Adding PolynomialsAdding PolynomialsTopic
6.1.2
The simple way to multiply a polynomial by a number is to multiply each term of the polynomial by the number.
In other words, you multiply out the parentheses, using the distributive property of multiplication over addition.
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Adding PolynomialsAdding Polynomials
Example 4
Topic
6.1.2
Multiply x2 + 2x – 4 by 3.
Solution
Solution follows…
3(x2 + 2x – 4) = (3 × x2) + (3 × 2x) – (3 × 4)
= 3x2 + 6x – 12
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Add these polynomials and simplify the resulting expressions.
13. (4x2 – 2x – 1) + (3x2 + x – 10)
14. (11x4 – 5x3 – 2x) + (–7x4 + 3x3 + 5x – 3)
Lesson
1.1.1
Guided Practice
Adding PolynomialsAdding PolynomialsTopic
6.1.2
Solution follows…
16. 5x2 + 3x – 3 –4x2 – 3x + 5 –2x2 + x – 7
(4x2 + 3x2) + (–2x + x) + (–1 – 10)= 7x2 – x – 11
c3 + c2 + 2c + 2 –x2 + x – 5
15. –5c3 – 3c2 + 2c + 1 4c2 – c – 3 6c3 + c + 4
(11x4 – 7x4) + (–5x3 + 3x3) + (–2x + 5x) + (0 – 3)= 4x4 – 2x3 + 3x – 3
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Lesson
1.1.1
Guided Practice
Adding PolynomialsAdding PolynomialsTopic
6.1.2
Solution follows…
Multiply these polynomials by 4.
17. 10y2 – 7y + 5 18. (x2 – 3x + 3)
19. (–x2 + x – 4) 20. (2x2 + 5x + 2)
4(10y2) – 4(7y) + 4(5)= 40y2 – 28y + 20
4(x2) – 4(3x) + 4(3)= 4x2 – 12x + 12
4(–x2) + 4(x) – 4(4)= –4x2 + 4x – 16
4(2x2) + 4(5x) + 4(2)= 8x2 + 20x + 8
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In Exercises 1-5, simplify the expression and state the degree of the resulting polynomial.
Adding PolynomialsAdding Polynomials
Independent Practice
Solution follows…
Topic
6.1.2
1. (2x2 + 3x – 7) + (7x2 – 3x + 4)
2. (x3 + x – 4) + (x3 – 8) + (4x3 – 3x – 1)
3. (–x6 + x – 5) + (2x6 – 4x – 6) + (–2x6 + 2x – 4)
4. (3x2 – 2x + 7) + (4x2 + 6x – 8) + (–5x2 + 4x – 5)
5. (0.4x3 – 1.1) + (0.3x3 + x – 1.0) + (1.1x3 + 2.1x – 2.0)
9x2 – 3, degree 2
6x3 – 2x – 13, degree 3
–x6 – x – 15, degree 6
2x2 + 8x – 6, degree 2
1.8x3 + 3.1x – 4.1, degree 3
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In Exercises 6-7, simplify the expression and state the degree of the resulting polynomial.
Independent Practice
Solution follows…
Topic
6.1.2
6. – 4a3 – 2a + 38a4 – 2a3 – 4a + 87a4 – 4a – 7
7. 1.1c2 + 1.4c – 0.48–4.9c2 – 3.6c + 0.987.3c2 + 0.13
15a4 – 6a3 – 10a + 4, degree 4 3.5c2 – 2.2c + 0.63, degree 2
Adding PolynomialsAdding Polynomials
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Independent Practice
Solution follows…
Topic
6.1.2
Multiply each polynomial below by –4.
8. 4a2 + 3a – 2
9. –c2 + 3c + 1
10. –6x3 – 4x2 + x – 8
11. 24x3 + 16x2 – 4x + 32
–16a2 – 12a + 8
4c2 – 12c – 4
24x3 + 16x2 –4x + 32
–96x3 – 64x2 + 16x – 128
Adding PolynomialsAdding Polynomials
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Independent Practice
Solution follows…
Topic
6.1.2
Multiply each polynomial below by 2a.
12. 3a2 + a – 8
13. –7a4 + 2a2 – 5a + 4
6a3 + 2a2 – 16a
–14a5 + 4a3 – 10a2 + 8a
Adding PolynomialsAdding Polynomials
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Topic
6.1.2
Round UpRound Up
Adding polynomials can look hard because there can be several terms in each polynomial.
The important thing is to combine each set of like terms, step by step.
Adding PolynomialsAdding Polynomials
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