XEI: Expressions, equations, and inequalities

Polynomials What are polynomials? Polynomials are algebraic expressions defined by its degree and number of terms. What are terms? Terms are parts of an expression separated by a “+” or “-” sign.

Examples of polynomials and terms Example 1 – 2x² - 3x + 4 is a polynomial 2x², -3x, and 4 are all terms So, this polynomial has 3 terms Example 2 – x – 9 is a polynomial x and -9 are terms So, this polynomial has 2 terms

Degrees of Polynomials What is the degree of a polynomial? The degree of a polynomial is the highest power of the variable in the polynomial. Example 1 – 2x² - 3x + 4 is a polynomial x² is the exponent with the highest power in the polynomial So, the degree of the polynomial is 2

More Examples of Degrees of Polynomials Example 3 – 4x³ - 2x² + 5x – 7 is a polynomial x³ is the exponent with highest power So, the degree of the polynomial is 3 Example 4 – 10 is a polynomial It has no variable, so its degree is 0

Naming Polynomials How are polynomials named? Polynomials are named according to their degree and number of terms

Polynomial Naming Chart Degree Terms Constant n/a 1 Linear Monomial 2 Quadratic Binomial 3 Cubic Trinomial 4 Quartic Polynomial w/ 4 terms 5 Quintic Polynomial w/ 5 terms

Examples of Naming Polynomials Example 1 – 2x² - 3x + 4 has a degree of 2 and has 3 terms So, it is a quadratic trinomial Example 2 – x – 9 has a degree of 1 and has 2 terms So, it is a linear binomial

More Examples Example 3 – 4x³ - 2x² + 5x – 7 has a degree of 3 and has 4 terms So, it is a cubic polynomial with 4 terms Example 4 – 10 has a degree of 0 and has 1 term So, it is a constant monomial or often referred to as just a “constant”

You Try Name the following polynomials: 1. 8x³ - 27 2. 2x² - 7x + 10 3. 5x 4. -3x⁴ - 2x² + 1

HOMEWORK Do all of the problems on the “XEI: Naming Polynomials” sheet

Multiplying Polynomials What are like terms? Like terms are terms that have the same variable and power Like terms can be added/subtracted. This is what it means to combine like terms.

How To Use An Expansion Box Monomial x Binomial Example 1: a(b + c) 1. Put the first factor on the side and the other factor on the top 2. Add what is in the boxes. Combine all like terms is possible. So, a(b + c) = ab + ac b +c a a ∙ b a ∙ c

More Examples Of Using An Expansion Box Binomial x Binomial Example 2: (a + b)(c + d) = 1. Put first factor on side and other factor on top. 2. Add the boxes and combine like terms if possible. ac + ad + bc + bd c +d a a ∙ c a ∙ d +b b ∙ c b ∙ d

Real Examples Example 3 – 7(x + 2) = 7x + 14 x +2 7 7x 14

More Real Examples Example 4 – (x + 6)(x – 4) = x² -4x + 6x – 24 = x² + 2x – 24 x -4 x² -4x +6 6x -24

You Try 2x(3x + 7) (x – 5)(4x + 2) 3x²(2x² - 3x + 4)

Homework Do all of the problems on “XEI: Multiplying Radicals”

Factoring Simple Quadratics What is a quadratic expression? A quadratic expression is a single variable, degree 2 polynomial Examples of quadratic expressions: 1. x² + 10x + 21 2. x² - 81 3. 4x² - 25 4. x² - 7x 5. -6x² - 22x -20

Simple Quadratic Expressions What are simple quadratic expressions? Simple quadratic expressions are in the form x² + bx + c How do we factor simple quadratic expressions? 1. Find the pair of numbers that multiply to produce c and add up to b 2. Place those two numbers in the following binomial factors: (x + __)(x + __)

Examples of Factoring Simple Quadratics Example 1 – Factor x² + 10x + 21 1. Find the pair of factors of 21 that add up to 10: 1, 21  1 + 21 = 22  NO -1, -21  -1 + -21 = -22  NO 3, 7  3 + 7 = 10  YES -3, -7  -3 + -7 = -10  NO 2. So, x² + 10x + 21 = (x + 3)(x + 7)

More Examples of Factoring Quadratics Example 2 – Factor x² + 7x – 30 1. Find the pair of factors of -30 that add up to 7: 1, -30  1 + -30 = -29  NO -1, 30  -1 + 30 = 29  NO 5, -6  5 + -6 = -1  NO -5, 6  -5 + 6 = 1  NO 3, -10  3 + -10 = -7  NO -3, 10  -3 + 10 = 7  YES 2. So, x² + 7x – 30 = (x + -3)(x + 10)

More Examples of Factoring Quadratics Example 3 – Factor x² -14x + 48 1. Find the pair of factors of 48 that add up to -14: 1,48  1 + 48 = 49  NO -1,-48  -1 + -48 = -49  NO 2, 24  2 + 24 = 26  NO -2,-24  -2 + -24 = -26  NO 3, 16  3 + 16 = 19  NO -3,-16  -3 + -16 = -19  NO 4, 12  4 + 12 = 16  NO -4,-12  -4 + -12 = -16  NO 6, 8  6 + 8 = 14  NO -6,-8  -6 + -8 = -14  YES 2. So, x² - 14x + 48 = (x + -6)(x + -8)

YOU TRY x² + 5x – 24 x² - 8x – 33 x² + 15x + 50

Homework Do all of the problems on “XEI: Factoring Quadratics”

Factor Other Quadratic Expressions What are perfect square trinomials? Perfect square trinomials are produced by one binomial multiplied by itself In x² + bx + c, c is the square of a number and b is 2 times that number. Example 1 – x² + 16x + 64 = (x + 8)(x + 8) = (x + 8)² since 64 is 8² and 16 is 2 ∙ 8

More Examples of Perfect Square Trinomials Example 2 – x² - 6x + 9 = (x + -3)(x + -3) = (x + -3)² since 9 is (-3)² and -6 is 2 ∙ -3 Example 3 – x² + 12x + 36 = (x + 6)(x + 6) = (x + 6)² Example 4 – x² -20x + 100 = (x + -10)(x + -10) = (x + -10)²

You Try Factor the following quadratics: x² - 24x + 144 x² + 10x + 25

Difference of Squares What is difference of squares? Difference of squares is a binomial in which one square is being subtracted from another. Factors of difference of squares are two binomials where one is the sum of the square roots of the squares and the other is the difference of the square roots of the squares.

Examples of Difference of Squares Example 1 – x² - 81 = (x + 9)(x – 9) since x² and 81 are perfect squares Example 2 – x² - 100 = (x + 10)(x – 10) since x² and 100 are perfect squares

You Try Factor the following quadratics: x² - 25 9x² - 36

HOMEWORK Do all of the problems on “XEI: Factoring Special Quadratics”