The following topics are covered on the test: 1. Polynomials. 1. Name polynomials by degree and by the number of terms. 2. Know the nicknames for zero-th degree (constant) first degree (linear), second degree (quadratic) and third degree (cubic) polynomials 3. Identify the leading coefficient. 2. Adding polynomials. Subtracting polynomials. 3. Multiplying polynomials. 1. One term by many terms. (Distribution) 2. Two terms by two terms. (FOIL) 3. Two terms by three terms.
4. Factoring quadratic polynomials. 1. Factoring out the greatest common factor. 2. Factoring quadratic trinomials: easy and difficult cases. 3. Factoring out the negative sign and greatest common factor and then factoring. 5. Polynomial Equations 1. Determine the degree of a polynomial equation. (Know nicknames for first degree, second degree and third degree) 2. Determine up to how many solutions can the polynomial equation have. 3. Determine if a given value of x is a solution or not.
6. Solving quadratic equations. 1. Solving quadratic equations with linear term missing. 2. Solving quadratic equations with the constant term missing. 3. Solving quadratic equations with no terms missing.
Ex 1: Given polynomials, determine the degree, the number of terms and the leading coefficient.
Ex 2: Given a polynomial, classify it as a constant, a linear, a quadratic or a cubic polynomial.
Ex 3: Add or subtract the polynomials below. a) ( x³ + 6x² - 2) + ( -2x³ + x ² +10) = x³ + 6x² x³ + x ² +10 = - x³ + 7x² + 8 b ) ( x³ + 6x² - 2) - ( -2x³ + x ² +10) = x³ + 6x² x³ - x ² - 10 = 3x³ + 5x² - 12
Recall: If there is nothing in front of the parentheses, then you can just drop them. If there is something directly in front of the parentheses like a negative sign, you must distribute it first before you can drop the parentheses.
Ex 3: In the examples that follow, multiply the following polynomials. a) 2x² ( x³ - 5x² +10) = 2x⁵ - 10 x ⁴ + 20x²
Try a few FOIL problems together.
Let’s try one square of a binomial problem: Let’s try one sum and difference pattern problem:
Let’s try a few problems from the homework together. 1) 2)
If there is a term in front of the FOIL problem, you must perform the FOIL first and then distribute the term. 3)
Let’s try one Polynomial by a Polynomial Problem.
Ex 4: In the example that follow, factor the following polynomials. a) Factor out the greatest common factor. 10x² + 20x = 10x ( x + 2) b) Factor this polynomial. x² - 5x + 6 = (x - 2 )(x- 3 ) c) Factor this polynomial. 2x² + 9x – 5 = (2x-1 )(x +5 ) d) Factor this polynomial. (Remember to factor out the negative sign first) -x² - 2x + 80 = -(x² +2x -80) = - (x +10)(x-8)
Ex 5: Classify each polynomial equation according to its degree. Comment on the number of solutions you would expect. 1) 2x + 6 = 3 This is a first degree or a linear equation. It can have at most one solution. 2) 9x² - 35 = 14 This is second degree or a quadratic equation. It can have up to two solutions.
3)15x² = 0 This is second degree or a quadratic equation. It can have up to two solutions. 4) 10x³ + 8 = 1 This is a third degree or a cubic equation. It can have up to three solutions. 5) x² + 8 = 3 This is second degree or a quadratic equation. It can have up to two solutions.
Consider the Polynomial Equation Is x = 0 a solution? Is x = 2 a solution? YES. Is x = -1 a solution? YES.
Consider this quadratic equation: 2x² = 50 1) How many solutions do you expect? We expect up to two solutions. 2) Is x = 5 a solution? Yes. 2(5) ² = 50 2*25 = = 50 3) Is x = -5 a solution? Yes. 2(-5) ² = 50 2*25 = = 50
Consider this quadratic equation: x² + 2x+1 = 0 1) How many solutions do you expect? We expect up to two solutions. 2) Is x = 0 a solution? No. (0)² + 2(0)+1 = = 0 1 = 0 3) Is x = 1 a solution? NO(1)² + 2(1)+1 = = 0 4 = 0
Consider this quadratic equation: x² + 2x+1 = 0 1) Is x = -1 a solution? YES. (-1)² + 2(-1)+1 = = 0 0 = 0
Given a quadratic equation, determine which term is missing. 1) x² - 7x + 12 = 0 No terms are missing. 2) x² = 100 The first degree (linear) term is missing. 3) x² + 10 = 26 The first degree (linear) term is missing. 4) 2x²+12x = 0 The constant term is missing.
Given a quadratic equation, determine which term is missing. 5) -6 x² + 21x – 18 = 0 No terms are missing. 6) 3 x² = - 6x The constant term is missing. 7) 3 x² + 12 = 12x No terms are missing.
Given a quadratic equation, which term is missing? 1) x² - 9 = 16 The linear term is missing. 2) 45x² - 3x = 0 The constant term is missing. 3) 8x² - 20x = 0 The constant term is missing. 4) 4x² = 36 The linear term is missing.
Given a quadratic equation, which term is missing? 5) x² + 9 = 45 The linear term is missing. 6) 9x² = 49 The linear term is missing. 7) x² - 4x + 4 = 0 No terms are missing. 8) x² = - 4x – 6 No terms are missing.
Recall: Depending on which term is missing from the quadratic equation, will tell you which method to use to solve it. In the upcoming examples, see if you can solve these quadratic equations.
(No terms are missing) x² + x – 20 = 0 (x + 5)(x – 4) = 0 x + 5 = 0 x – 4 = 0 x = - 5 x = 4
(No terms are missing) x² - 7x = - 6 x² - 7x + 6 = 0 (x - 1)(x – 6) = 0 x - 1 = 0 x – 6 = 0 x = 1 x = 6
Solve this quadratic equation. -2x² - 13x = x² - 13x + 7 = 0 - (2x² + 13x – 7) = 0 - (2x - 1)(x + 7 ) = 0 2x - 1 = 0 x + 7 = 0 2x = 1 x = - 7 x= 1/2
Solve this quadratic equation. (No missing terms) 2x² - 48x = x² - 48x + 288= 0 2(x² - 24x + 144) = 0 2(x-12) (x-12) = 0 x- 12 = 0x – 12 = 0 x = 12
Solve this quadratic equation. -3x² + 21x = 18 -3x² + 21x - 18 = 0 -(3x² - 21x + 18) = 0 -3(x² - 7x + 6 ) = 0 -3(x - 1)(x – 6) = 0 x - 1 = 0 x – 6 = 0 x = 1 x = 6
Solve this quadratic equation. (Missing Constant Term) 3x² = x 3x² - x = 0 x(3x – 1 ) = 0 x = 0 3x-1 = 0 3x = 1 x = 1/3
Solve this quadratic equation. (Missing Linear Term) 3x² + 6 = 33 3x² = 27 x²= 9 x = 3, x = -3
Solve this quadratic equation. (Missing Linear Term)
Solve this quadratic equation. (No terms missing) x² + x – 20 = 0 (x + 5)(x – 4) = 0 x + 5 = 0 x – 4 = 0 x = - 5 x = 4
Solve this quadratic equation. (Missing Constant Term) 3x² = x 3x² - x = 0 x(3x – 1 ) = 0 x = 0 3x-1 = 0 3x = 1 x = 1/3