Adding, Subtracting and Multiplying Polynomials
Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required)
To access a customizable version of this book, as well as other interactive content, visit www.ck12.org
CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-source, collaborative, and web-based compilation model, CK-12 pioneers and promotes the creation and distribution of high-quality, adaptive online textbooks that can be mixed, modified and printed (i.e., the FlexBook® textbooks). Copyright © 2015 CK-12 Foundation, www.ck12.org The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License (http://creativecommons.org/ licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/about/ terms-of-use. Printed: July 8, 2015
www.ck12.org
C HAPTER
Chapter 1. Adding, Subtracting and Multiplying Polynomials
1
Adding, Subtracting and Multiplying Polynomials
Objective To add, subtract, and multiply polynomials. Review Queue 1. Multiply (3x − 1)(x + 4). 2. Factor x2 − 6x + 9. 3. Multiply (2x + 5)(2x − 11). 4. Combine like terms: 3x + 15 + 2x − 8 − x
Adding and Subtracting Polynomials Objective Adding and subtracting polynomials, as well as learning about the different parts of a polynomial. Watch This
MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/60114
James Sousa: Ex: Intro to Polynomials in One Variable Guidance A polynomial is an expression with multiple variable terms, such that the exponents are greater than or equal to zero. All quadratic and linear equations are polynomials. Equations with negative exponents, square roots, or variables in the denominator are not polynomials.
Now that we have established what a polynomial is, there are a few important parts. Just like with a quadratic, a polynomial can have a constant, which is a number without a variable. The degree of a polynomial is the largest 1
www.ck12.org exponent. For example, all quadratic equations have a degree of 2. Lastly, the leading coefficient is the coefficient in front of the variable with the degree. In the polynomial 4x4 + 5x3 − 8x2 + 12x + 24 above, the degree is 4 and the leading coefficient is also 4. Make sure that when finding the degree and leading coefficient you have the polynomial in standard form. Standard form lists all the variables in order, from greatest to least. Example A Rewrite x3 − 5x2 + 12x4 + 15 − 8x in standard form and find the degree and leading coefficient. Solution: To rewrite in standard form, put each term in order, from greatest to least, according to the exponent. Always write the constant last.
x3 − 5x2 + 12x4 + 15 − 8x → 12x4 + x3 − 5x2 − 8x + 15 Now, it is easy to see the leading coefficient, 12, and the degree, 4. Example B Simplify (4x3 − 2x2 + 4x + 15) + (x4 − 8x3 − 9x − 6) Solution: To add or subtract two polynomials, combine like terms. Like terms are any terms where the exponents of the variable are the same. We will regroup the polynomial to show the like terms.
(4x3 − 2x2 + 4x + 15) + (x4 − 8x3 − 9x − 6) x4 + (4x3 − 8x3 ) − 2x2 + (4x − 9x) + (15 − 6) x4 − 4x3 − 2x2 − 5x + 9 Example C Simplify (2x3 + x2 − 6x − 7) − (5x3 − 3x2 + 10x − 12) Solution: When subtracting, distribute the negative sign to every term in the second polynomial, then combine like terms.
(2x3 + x2 − 6x − 7) − (5x3 − 3x2 + 10x − 12) 2x3 + x2 − 6x − 7 − 5x3 + 3x2 − 10x + 12 (2x3 − 5x3 ) + (x2 + 3x2 ) + (−6x − 10x) + (−7 + 12) − 3x3 + 4x2 − 16x + 5 Guided Practice p 1. Is 2x3 − 5x + 6 a polynomial? Why or why not? 2. Find the leading coefficient and degree of 6x2 − 3x5 + 16x4 + 10x − 24. Add or subtract. 3. (9x2 + 4x3 − 15x + 22) + (6x3 − 4x2 + 8x − 14) 4. (7x3 + 20x − 3) − (x3 − 2x2 + 14x − 18) Answers 1. No, this is not a polynomial because x is under a square root in the equation. 2
www.ck12.org
Chapter 1. Adding, Subtracting and Multiplying Polynomials
2. In standard form, this polynomial is −3x5 + 16x4 + 6x2 + 10x − 24. Therefore, the degree is 5 and the leading coefficient is -3. 3. (9x2 + 4x3 − 15x + 22) + (6x3 − 4x2 + 8x − 14) = 10x3 + 5x2 − 7x + 8 4. (7x3 + 20x − 3) − (x3 − 2x2 + 14x − 18) = 6x3 + 2x2 + 6x + 15 Vocabulary Polynomial An expression with multiple variable terms, such that the exponents are greater than or equal to zero. Constant A number without a variable in a mathematical expression. Degree(of a polynomial) The largest exponent in a polynomial. Leading coefficient The coefficient in front of the variable with the degree. Standard form Lists all the variables in order, from greatest to least. Like terms Any terms where the exponents of the variable are the same. Problem Set Determine if the following expressions are polynomials. If not, state why. If so, write in standard form and find the degree and leading coefficient. 1. 2. 3. 4. 5. 6. 7. 8.
1 +x+5 x2 x3 + 8x4 − 15x + 14x2 − 20
x3 + 8 −1 5x−2 √ + 9x √ + 16 x2 2 − x 6 + 10
x4 +8x2 +12 3 x2 −4 x −6x3 + 7x5 − 10x6 + 19x2 − 3x + 41
Add or subtract the following polynomials. 9. 10. 11. 12. 13. 14. 15.
(x3 + 8x2 − 15x + 11) + (3x3 − 5x2 − 4x + 9) (−2x4 + x3 + 12x2 + 6x − 18) − (4x4 − 7x3 + 14x2 + 18x − 25) (10x3 − x2 + 6x + 3) + (x4 − 3x3 + 8x2 − 9x + 16) (7x3 − 2x2 + 4x − 5) − (6x4 + 10x3 + x2 + 4x − 1) (15x2 + x − 27) + (3x3 − 12x + 16) (2x5 − 3x4 + 21x2 + 11x − 32) − (x4 − 3x3 − 9x2 + 14x − 15) (8x3 − 13x2 + 24) − (x3 + 4x2 − 2x + 17) + (5x2 + 18x − 19) 3
www.ck12.org
Multiplying Polynomials Objective To multiply together several different types of polynomials. Watch This
MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/60115
James Sousa: Ex: Polynomial Multiplication Involving Binomials and Trinomials Guidance Multiplying together polynomials is very similar to multiplying together factors. You can FOIL or we will also present an alternative method. When multiplying together polynomials, you will need to use the properties of exponents, primarily the Product Property (am · an = am+n ) and combine like terms. Example A Find the product of (x2 − 5)(x3 + 2x − 9). Solution: Using the FOIL method, you need be careful. First, take the x2 in the first polynomial and multiply it by every term in the second polynomial.
Now, multiply the -5 and multiply it by every term in the second polynomial.
Lastly, combine any like terms. In this example, only the x3 terms can be combined.
Example B Multiply (x2 + 4x − 7)(x3 − 8x2 + 6x − 11). 4
www.ck12.org
Chapter 1. Adding, Subtracting and Multiplying Polynomials
Solution: In this example, we will use the “box” method. Align the two polynomials along the top and left side of a rectangle and make a row or column for each term. Write the polynomial with more terms along the top of the rectangle.
Multiply each term together and fill in the corresponding spot.
Finally, combine like terms. The final answer is x5 − 4x4 − 33x3 + 69x2 − 86x + 77. This method presents an alternative way to organize the terms. Use whichever method you are more comfortable with. Keep in mind, no matter which method you use, you will multiply every term in the first polynomial by every term in the second. Example C Find the product of (x − 5)(2x + 3)(x2 + 4). Solution: In this example we have three binomials. When multiplying three polynomials, start by multiplying the first two binomials together.
(x − 5)(2x + 3) = 2x2 + 3x − 10x − 15 = 2x2 − 7x − 15 Now, multiply the answer by the last binomial.
(2x2 − 7x − 15)(x2 + 4) = 2x4 + 8x2 − 7x3 − 28x − 15x2 − 60 = 2x4 − 7x3 − 7x2 − 28x − 60 Guided Practice Find the product of the polynomials. 1. −2x2 (3x3 − 4x2 + 12x − 9) 2. (4x2 − 6x + 11)(−3x3 + x2 + 8x − 10) 3. (x2 − 1)(3x − 4)(3x + 4) 5
www.ck12.org 4. (2x − 7)2 Answers 1. Use the distributive property to multiply −2x2 by the polynomial.
−2x2 (3x3 − 4x2 + 12x − 9) = −6x5 + 8x4 − 24x3 + 18x2 2. Multiply each term in the first polynomial by each one in the second polynomial.
(4x2 − 6x + 11)(−3x3 + x2 + 8x − 10) = −12x5 + 4x4 + 32x3 − 40x2 + 18x4 − 6x3 − 48x2 + 60x − 33x3 + 11x2 + 88x − 110 = −12x5 + 22x4 − 7x3 − 77x2 + 148x − 110 3. Multiply the first two binomials together.
(x2 − 1)(3x − 4) = 3x3 − 4x2 − 3x + 4 Multiply this product by the last binomial.
(3x3 − 4x2 − 3x + 4)(3x + 4) = 9x4 + 12x3 − 12x3 − 16x2 − 9x2 − 12x + 12x − 16 = 9x4 − 25x2 − 16 4. The square indicates that there are two binomials. Expand this and multiply.
(2x − 7)2 = (2x − 7)(2x − 7) = 4x2 − 14x − 14x + 49 = 4x2 − 28x + 49 Problem Set Find the product. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 6
5x(x2 − 6x + 8) −x2 (8x3 − 11x + 20) 7x3 (3x3 − x2 + 16x + 10) (x2 + 4)(x − 5) (3x2 − 4)(2x − 7) (9 − x2 )(x + 2) (x2 + 1)(x2 − 2x − 1) (5x − 1)(x3 + 8x − 12) (x2 − 6x − 7)(3x2 − 7x + 15) (x − 1)(2x − 5)(x + 8)
www.ck12.org 11. 12. 13. 14. 15. 16. 17.
Chapter 1. Adding, Subtracting and Multiplying Polynomials
(2x2 + 5)(x2 − 2)(x + 4) (5x − 12)2 −x4 (2x + 11)(3x2 − 1) (4x + 9)2 (4x3 − x2 − 3)(2x2 − x + 6) (2x3 − 6x2 + x + 7)(5x2 + 2x − 4) (x3 + x2 − 4x + 15)(x2 − 5x − 6)
7
