Topic 7: Polynomials

Table of Contents 1. Introduction to Polynomials 2. Adding & Subtracting Polynomials 3. Multiplying Polynomials 4. Factoring Polynomials 5. Factoring Polynomials, part 2 6. Solving Quadratics with Factoring 7. Factoring by Grouping 8. Completing the Square

Introduction to Polynomials

Vocab Monomial: a number, a variable, or a product of numbers and variables with whole-number exponents. Degree of a monomial: is the sum of the exponents of the variables. A constant has degree 0.

Example: Degree of a Monomial Find the degree of each monomial. A. 4p4q3 B. 7ed C. 3

Let’s Practice…. Find the degree of each monomial. a. 1.5k2m b. 4x c.

Review: Like Terms 4a3b2 + 3a2b3 – 2a3b2 You can add or subtract monomials by adding or subtracting like terms. Like terms The variables have the same powers. 4a3b2 + 3a2b3 – 2a3b2 Not like terms The variables have different powers.

Identify Like Terms Identify the like terms in each polynomial. A. 5x3 + y2 + 2 – 6y2 + 4x3 B. 3a3b2 + 3a2b3 + 2a3b2 – a3b2 Like terms: ______________________ Like terms: _______________________

Let’s Practice… Identify the like terms in each polynomial. A. 4y4 + y2 + 2 – 8y2 + 2y4 B. 7n4r2 + 3n2r3 + 5n4r2 + n4r2 Like terms: ____________________________ Like terms: ___________________________

Add or Subtract Monomials Simplify. A. 4x2 + 2x2

Add or Subtract Monomials Simplify. B. 3n5m4 - n5m4

Let’s Practice… Simplify. 2x3 - 5x3 2n5p4 + n5p4

Vocab Polynomial: an expression of more than two algebraic terms. Example: 3x4 + 5x2 – 7x + 1 Degree of a polynomial is the degree of the term with the greatest power/exponent. Example: The degree of 3x4 + 5x2 – 7x + 1 is 4.

Degree of a Polynomial A. 11x7 + 3x3 Find the degree of each polynomial. A. 11x7 + 3x3 B.

Let’s Practice… a. 5x – 6 b. x3y2 + x2y3 – x4 + 2 Find the degree of each polynomial. a. 5x – 6 b. x3y2 + x2y3 – x4 + 2

Vocab Standard form of a polynomial: Polynomial written with the terms in order from greatest degree to least degree. Leading Coefficient: When written in standard form, the coefficient of the first term is called the leading coefficient. Example: 3x4 + 5x2 – 7x + 1 and 3 is the leading coefficient.

Let’s Practice… 1. 6x – 7x5 + 4x2 + 9 2. 16 – 4x2 + x5 + 9x3 Write the polynomial in standard form. Then give the leading coefficient. 1. 6x – 7x5 + 4x2 + 9 2. 16 – 4x2 + x5 + 9x3 3. 18y5 – 3y8 + 14y

Special Polynomial Names By Degree Degree Name By # of Terms 1 2 Constant Linear Quadratic 3 4 5 6 or more 6th,7th,degree and so on Cubic Quartic Quintic Name Terms Monomial Binomial Trinomial Polynomial 4 or more 1 2 3

Name the Following… A. 5n3 + 4n B. 4y6 – 5y3 + 2y – 9 C. –2x Classify each polynomial according to its degree and number of terms. A. 5n3 + 4n B. 4y6 – 5y3 + 2y – 9 C. –2x

Let’s Practice… a. x3 + x2 – x + 2 b. 6 c. –3y8 + 18y5 + 14y Classify each polynomial according to its degree and number of terms. a. x3 + x2 – x + 2 b. 6 c. –3y8 + 18y5 + 14y

Adding and Subtracting Polynomials

Adding and Subtracting Polynomials Just as you can perform operations on numbers, you can perform operations on polynomials. To add or subtract polynomials, combine like terms. Remember! Like terms are constants or terms with the same variable(s) raised to the same power(s).

Simplifying Polynomials Combine like terms. A. 12p3 + 11p2 + 8p3 B. 5x2 – 6 – 3x + 8

Let’s Practice… Combine like terms. a. 2s2 + 3s2 + s – 3s2 – 5s b. 4z4 – 8 + -2z2 +16z4 + 2 + 5z3 – 7

Let’s Practice… Combine like terms. c. 2x8 + 7y8 – x8 – 9y7 - 10y7 + y8 d. 9b3c2 + -4b3 + 5c2 + 5b3c2 – 13b3c2

2 Methods: Adding Polynomials Polynomials can be added in either vertical or horizontal form. In vertical form, align the like terms and add: In horizontal form, use the Associative and Commutative Properties to regroup and combine like terms. 5x2 + 4x + 1 + 2x2 + 5x + 2 7x2 + 9x + 3 (5x2 + 4x + 1) + (2x2 + 5x + 2) = (5x2 + 2x2 + 1) + (4x + 5x) + (1 + 2) = 7x2 + 9x + 3

Adding Polynomials Add. A. (4m2 + 5) + (m2 – m + 6) B. (10xy + x) + (–3xy + y)

Let’s Practice… Add. Add (5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a).

Subtracting Polynomials To subtract polynomials, remember that subtracting is the same as adding the opposite (distributing the negative). To find the opposite of a polynomial, you must write the opposite of each term in the polynomial: –(2x3 – 3x + 7)= –2x3 + 3x – 7

Subtracting Polynomials (–10x2 – 3x + 7) – (x2 – 9)

Subtracting Polynomials (x3 + 4y) – (2x3) (7m4 – 2m2) – (5m4 – 5m2 + 8)

Let’s Practice… Subtract. (9q2 – 3q) – (q2 – 5) (2x2 – 3x2 + 1) – (x2 + x + 1)

Multiplying Polynomials F.O.I.L

Multiplying Polynomials Each term in the first polynomial, must be multiplied by each term in the second polynomial.

Method 1: Distribute First Outer Inner Last Multiply!!!

“F.O.I.L.”

Method 2: Box 3x -5 5x +2 Multiply (3x – 5)(5x + 2) Draw a box. Write a polynomial on the top and side of a box. Multiply. Combine like terms. 3x -5 5x +2

Let’s Practice… 1. (7x – 10)(3x + 8) 2. (2x – 3)(4x - 8)

Multiplying Terms with Exponents When FOILing, add the exponents and multiply coefficients. Add the little numbers and multiply the big numbers!!! Example: (3x2 + 10x)(5x3 – 7x2) 15x5 - 21x4 + 50x4 – 70x3 15x5 + 29x4 – 70x3

Let’s Practice… 1. (7x2 – 10x)(3x3 + 8x2) 2. (2x4 – 3x2)(4x - 8)

Multiplying Larger Polynomials Each term in the 1st polynomial must be multiplied by each term in the 2nd. Example: (7x2 + 2x + 8)(4x3 – 9x2)

Method 2: Multiply: (2x - 5)(x2 - 5x + 4)

Let’s Practice… (5x2 + 7) (2x3 – 5x2 +9) (10x4 – 5x2 + 8) (8x3 -3x -6)

Factoring Polynomials

Vocab: Factoring Factoring is rewriting an expression as a product of factors. It is the reverse of multiplying polynomials FOILing.

To determine the factors, ask yourself… What two #’s add to the middle number AND multiply to the last number?!?!

Let’s Practice… What adds (or subtracts) to get 3 and multiplies to get 2? What adds (or subtracts) to get -7 and multiplies to get 10? What adds (or subtracts) to get -7 and multiplies to get -44?

Let’s Practice… Factor: x2 + 5x + 6 2. x2 -7x + 10 3. x2 -11x +24

Signs of Factors b c Factors + +,+ - +,- (The factor w/ the greater absolute value is -) +,- (The factor w/ the greater absolute value is +) -, -

Vocab: GCF The greatest common factor (GCF) is a common factor of the terms in the expression. Example:

Vocab: Prime If a polynomials is “prime” it means there are no factors. Find the prime polynomials below: 1. x2 + 7x + 9 2. x2 + 5x + 4 3. x2 + 9x + 10

Let’s Practice…. Factor. y2 -10y +16 2. r2 -11r +24 3. n2 -15n +56 4. v2 + 5v -36

Let’s Practice… x2 + 12x + 36 x2 - 8x + 16 -x2 +11x -18 16- x2

Factoring Polynomials, Part 2

Expanded Form Expanded Form When factoring problems where a ≠ 1, we first want to get the problem into expanded form before we try to factor.

Creating Expanded Form Step 1: Multiply a·c Step 2: To get to expanded form ask yourself “What multiplies to get a·c, and add/subtracts to get to b.” Example: Expand: 2x2 +9x +7 Expand: 3x2 + 2x – 8

Method 1: Step 3: Write your new factors in place of bx. Step 4: Group the first two terms together and the last two terms together. Step 5: Factor each group Step 6: Factor again to get the complete factorization

Method 1: 6x2 + 13 x +5 Multiply a·c (6·5=30) To get to expanded form ask yourself “What multiplies to get a·c, and add/subtracts to get to b.” (10, 3) Write your new factors in place of bx. (6x2+10x+3x+5) Group the first two terms together and the last two terms together. [(6x2+10x)+(3x+5)] Factor each group [2x(3x+5)+1(3x+5)] Factor again to get the complete factorization [(3x+5)(2x+1)]

Method 2: 6x2 + 13 x +5 Original 1st Term Expanded Term 1 Expanded Term 2 Original Last Term Step 3: Fill in box. Step 4: Factor horizontally and vertically. Step 5: Terms outside of box are the solution.

Factor: 2x2 + 5x -12 Original 1st Term Expanded Term 1 Expanded Term 2 Original Last Term

Factor: 3x2 + 7x +2

Factor: 2x2 + 15x -8

Factor: 16x2 + 28x +10

Factoring by Grouping

Factoring by Grouping Using the distributive property to factor polynomials with four or more terms. Terms can be put into groups and then factored---- each group will have a “like” factor used in regrouping.

Factoring by Grouping A polynomial can be factored by grouping if all of the following conditions exist. There are four or more terms. Terms have common factors that can be grouped together, and There are two common factors that are identical. Symbols: ax + bx + ay + by = (ax + bx) + (ay + by) Group, factor Regroup = x(a + b) + y(a + b) = (x + y)(a + b)

Factor by Grouping 6h4 – 4h3 + 12h – 8 Factor each polynomial by grouping. Check your answer. 6h4 – 4h3 + 12h – 8

Factor by Grouping 5y4 – 15y3 + y2 – 3y Factor each polynomial by grouping. 5y4 – 15y3 + y2 – 3y

Let’s Practice… Factor each polynomial by grouping. 6b3 + 8b2 + 9b + 12

Let’s Practice… Factor each polynomial by grouping. 4r3 + 24r + r2 + 6

Factoring with Opposite Groups 2x3 – 12x2 + 18 – 3x

Let’s Practice… 15x2 – 10x3 + 8x – 12 Factor each polynomial. Check your answer. 15x2 – 10x3 + 8x – 12

Let’s Practice… Factor each polynomial by grouping. 1. 2x3 + x2 – 6x – 3 2. 7p4 – 2p3 + 63p – 18

Factoring Procedure

Completing the Square

How to: Square root each side and solve. Rearrange equation so it is in the form ax2 + bx = c Divide every term on both sides by a. Add ( 𝑏 2 )2 to both sides of the equation. Factor. Square root each side and solve.

Solve: x2 + 6x - 3 = 0

Solve: x2 - 12x + 7 = 0

Solve: 2x2 + 2x -5 = x2

Solve: 9x2 - 12x - 2 = 0