Unit 7: Factoring Quadratic Polynomials A polynomial is represented by: where the coefficients are real numbers and the exponents are nonnegative integers. Side Note: Examples of real numbers: Examples of non negative integers: Example 1: Out of the following, circle the ones that are NOT polynomials. Give a reason for your answer. A. 5 5 3 B. 3 5 9 C. 3 D. 1 E. 4 3 F. 3 9 G. 6 4. 2 H. 13 8 I. Polynomials are named according to the number of terms they have. Name Number of Terms Examples Monomial 7 3 Binomial 12 2 3 5 Trinomial 2 9 10 10 7 4 Example 2: Name the parts of each polynomial. 8 10 5 Number of Terms: Leading Coefficient: Degree of the Polynomial: 8 9 4 3 Number of Terms: Leading Coefficient: Degree of the Polynomial:
Adding and Subtracting Polynomials To add polynomials, simply combine like terms. 18 12 4 5 25 To subtract polynomials, distribute the negative and then combine like terms. 3 5 6 Example 3: Explain what went wrong in Kamisa s and Marco s calculations. Then redo the problem correctly. Kamisa s Mistake Mistake: Marco s Mistake Mistake: 2 1 4 5 3 2 3 2 1 4 5 3 2 3 2 6 5 2 Correction: Correction: Example 4: Determine the sum or difference of each. Show all of your work. A. 4 5 2 2 7 8 9 B. 9 5 8 2 Multiplying by a Monomial To multiply by a monomial, use the distributive property. 3 2 Example 5: Determine each product by using the distributive property. A. 5 4 B. 2 4 7 C. 7 2 5 3 D. 4 5 3
Multiplying by a Binomial To multiply by a binomial, use the distributive property twice. 2 3 5 Example 6: Determine each product by using the distributive property twice. A. 4 3 B. 5 1 C. 2 3 2 D. 2 1 2 15 Example 7: More Practice binomial & binomial A. 3 12 4 6 B. 8 10 6 3 F O I L This process can be extended to higher order polynomials To multiply by a trinomial, the distributive property would be used 3 times. Example 8: Multiply the pair of trinomials by using the distributive property three times. 2 8 2 4 1 Example 9: Use a graphing calculator to verify the product from the worked example. Factored Form: 1 3 Standard Form: 2 3 Sketch both graphs on the coordinate plane. How do the graphs verify the 1 3 and 2 3 are equivalent functions? Which of these forms is better for finding the y intercept? Why? Which of these forms is better for finding the x intercepts? Why?
Example 1: Review Multiply each pair of polynomials A. 5 2 B. 2 9 C. 5 3 6 Example 2: Find the greatest common factor of each pair of expressions. A. 55 and 77 B. 56 and 24 C. 35 and 21 Example 3: Find the GCF of each polynomial, then un distribute it to factor out the greatest common factor A. 3 9 B. 20 10 12 C. 2 4 8 D. 12 24 E. 3 9 12 F. 45 63 An Introduction to Factoring Example 4: Let s Play a Game Find two numbers that satisfy the given conditions. A. Multiply to 15 and add to 8 B. Multiply to 20 and add to 12 C. Multiply to 20 and add to 9 D. Multiply to 32 and add to 4 E. Multiply to 63 and add to 2 F. Multiply to 27 and add to 12 Seeing the Connection between Standard Form and Factored Form Example 5: Graph 7 10 on your calculator. What are the zeros of the parabola? Write the equation in factored form: What do and multiply to? What do and add to? Do you see these numbers in the standard form of the function? Where?
Factoring Convert Standard Form into Factored Form Example 6: Write 9 20 in factored form, i.e. factor 9 20 What multiplies to and adds to? Answer: and Factored Form: Example 7: Factor 7 18 What multiplies to and adds to? Answer: and Factored Form: Example 8: If possible, factor each of the following expressions. Write PRIME if not factorable. A. 5 24 B. 3 28 C. 10 24 D. 13 36 E. 4 32 F. 6 18 Including a Greatest Common Factor Example 9: Completely factor each expression. Factor out the greatest common factor, then factor the remaining trinomial. A. 2 8 6 B. 2 12 80 C. 4 20 16 D. 2 4 30 E. 2 22 56 F. 5 25 250
Perfect Square Trinomials Example 10: Factor each expression. A. 6 9 B. 18 81 C. 2 28 98 What do you notice about each factored form? Difference of Squares Example 11: Determine each product. A. 3 3 B. 5 5 C. 2 7 2 7 What do you notice about each product? Example 12: Factor each expression. A. 4 B. 36 C. 4 9 D. 1 E. 16 81 F. 9 G. 2 2 H. 3 75 I. 4 98 Example 13: Factor each expression. A. 4 B. 81
Example 1: Review simplify each expression. A. 3 2 9 B. 3 2 5 C. 2 7 1 D. 1 4 3 Example 2: Review factor each expression. A. 5 6 B. 5 6 C. 3 12 36 Factoring by Grouping Example 3: 2 7 3 What if the leading coefficient isn t 1 and the gcf doesn t help? CHECK for greatest common factor MULTIPLY and. Find FACTORS of that add to. REWRITE using those factors of GROUP the first two terms and the last two terms. Find the GCF of each group. Parentheses become one factor and the GCFs become the other Example 4: 2 11 12 CHECK for greatest common factor MULTIPLY and. Find FACTORS of that add to. REWRITE using those factors of GROUP the first two terms and the last two terms. Find the GCF of each group. Parentheses become one factor and the GCFs become the other
Example 5: 8 16 6 CHECK for greatest common factor MULTIPLY and. Find FACTORS of that add to. REWRITE using those factors of GROUP the first two terms and the last two terms. Find the GCF of each group. Parentheses become one factor and the GCFs become the other Example 6: Factor each expression by grouping. Hint: Check for a greatest common factor first. A. 2 5 3 B. 3 10 8 C. 2 15 7 D. 7 11 4 E. 5 17 6 F. 8 10 3 G. 4 14 30 H. 18 21 15
All Mixed Up Example 7: Factor each expression. Hint: always check for a greatest common factor first. A. 2 18 B. 10 23 5 C. 6 D. 4 8 3 E. 6 14 6 F. 4 64 G. 2 2 40 H. 9 64 2 TERMS Greatest Common Factor? 3 TERMS Greatest Common Factor? You re done! Is it a Difference of Squares? If 1, find two numbers that multiply to and add to. Go straight to. If 1, find two numbers that multiply to and add to. Then factor by grouping.
Example 1: Review factor each expression. A. 3 15 18 B. 24 14 3 Quick Review: Give an example of a perfect square? What makes it a perfect square? Give an example of a perfect cube? What makes it a perfect cube? Sum and Difference of Cubes Example 2: How to factor something that is a perfect cube A trick for remembering the signs is S O A P A. 125 B. 125 C. 29 D. 1 E. 8 27 F. 27 125
Factoring 4 terms When we have four terms we can factor them by also using factoring by grouping. Example 3: Factor the following examples. A. 3 5 15 B. 1 C. 2 2 D. 4 12 5 15 E. 21 15 35 25 F. 12 4 15 5 If is a polynomial function and a is a real number, the following statements are equivalent: is a solution of the polynomial equation 0. is a zero of the function., 0 is an x intercept of the graph of. is a factor of. Example 4: 2 A. Factor B. Solve C. Find the zeros D. Graph the x intercepts