Section 5.1 Quadratic Equations

OBJECTIVES Find the greatest common factor (GCF) of numbers. Find the GCF of terms.

OBJECTIVES Factor out the GCF. D Factor a four-term expression by grouping.

Greatest Common Factor (GCF) DEFINITION Greatest Common Factor (GCF) The largest common factor of the integers in a list.

PROCEDURE Finding the Product 4(x + y) = 4x + 4y 5(a – 2b) = 5a – 10b 2x(x + 3) = 2x2 + 6x

PROCEDURE Finding the Factors 4x + 4y = 4(x + y) 5a – 10b = 5(a – 2b) 2x2 + 6x = 2x(x + 3)

DEFINITION GCF of a Polynomial a is the greatest integer that divides each coefficient.

DEFINITION GCF of a Polynomial n is the smallest exponent of x in all the terms.

Section 5.1 Exercise #2 Chapter 5 Factoring Let’s work Exercise #19 from Section 5.1

Section 5.1 Exercise #5 Chapter 5 Factoring Let’s work Exercise #19 from Section 5.1

Factor out GCF (3) Terms (2) Terms (4) Terms Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

Section 5.2 Quadratic Equations

OBJECTIVES A Factor trinomials of the form

RULE Factoring Rule 1

PROCEDURE Factoring x2 + bx + c Find two integers whose product is c and whose sum is b. If b and c are positive, both integers must be positive.

PROCEDURE Factoring x2 + bx + c Find two integers whose product is c and whose sum is b. If c is positive and b is negative, both integers must be negative.

PROCEDURE Factoring x2 + bx + c Find two integers whose product is c and whose sum is b. If c is negative, one integer must be negative and one positive.

Section 5.2 Exercise #6 Chapter 5 Factoring Let’s work Exercise #19 from Section 5.1

Factor out GCF (3) Terms (2) Terms (4) Terms Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

Section 5.3 Quadratic Equations

OBJECTIVES A Use the ac test to determine whether

OBJECTIVES B

OBJECTIVES C

TEST ac test for ax2 + bx + c A trinomial of the form ax2 + bx + c is factorable if there are two integers with product ac and sum b.

TEST ac test We need two numbers whose product is ac. The sum of the numbers must be b.

PROCEDURE Factoring by FOIL Product must be c. Product must be a.

PROCEDURE Factoring by FOIL The product of the numbers in the first (F) blanks must be a.

PROCEDURE Factoring by FOIL The coefficients of the outside (O) products and the inside (I) products must add up to b.

PROCEDURE Factoring by FOIL The product of numbers in the last (L) blanks must be c.

Section 5.3 Exercise #8 Chapter 5 Factoring Let’s work Exercise #19 from Section 5.1

Factor out GCF (3) Terms (2) Terms (4) Terms Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

Factor out GCF (3) Terms (2) Terms (4) Terms Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

Section 5.4 Quadratic Equations

OBJECTIVES A Recognize the square of a binomial (a perfect square trinomial).

OBJECTIVES B Factor a perfect square trinomial.

OBJECTIVES C Factor the difference of two squares.

RULES Factoring Rules 2 and 3: PERFECT SQUARE TRINOMIALS

RULES Factoring Rules 2 and 3: PERFECT SQUARE TRINOMIALS

RULE Factoring Rule 4: THE DIFFERENCE OF TWO SQUARES

Section 5.4 Exercise #11 Chapter 5 Factoring Let’s work Exercise #19 from Section 5.1

Factor out GCF (3) Terms (2) Terms (4) Terms Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

Section 5.4 Exercise #13 Chapter 5 Factoring Let’s work Exercise #19 from Section 5.1

Factor out GCF (3) Terms (2) Terms (4) Terms Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

Section 5.5 Quadratic Equations

OBJECTIVES A Factor the sum or difference of two cubes.

OBJECTIVES B Factor a polynomial by using the general factoring strategy.

OBJECTIVES C Factor expressions whose leading coefficient is –1.

RULE Factoring Rule 5: THE SUM OF TWO CUBES.

RULE Factoring Rule 6: THE DIFFERENCE OF TWO CUBES.

PROCEDURE General Factoring Strategy Factor out all common factors.

PROCEDURE General Factoring Strategy Look at the number of terms inside the parentheses. If there are: Four terms: Factor by grouping.

PROCEDURE General Factoring Strategy Three terms: If the expression is a perfect square trinomial, factor it. Otherwise, use the ac test to factor.

PROCEDURE General Factoring Strategy Two terms and squared: Look at the difference of two squares (X 2–A2) and factor it. Note: X 2+A2 is not factorable.

PROCEDURE General Factoring Strategy Two terms and cubed: Look for the sum of two cubes (X 3+A3) or the difference of two cubes (X 3-A3) and factor it.

PROCEDURE General Factoring Strategy Make sure your expression is completely factored. Check by multiplying the factors you obtain.

Section 5.5 Chapter 5 Factoring Let’s work Exercise #19 from Section 5.1

Section 5.5 Exercise #15 Chapter 5 Factoring Let’s work Exercise #19 from Section 5.1

Factor out GCF (3) Terms (2) Terms (4) Terms Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

Section 5.5 Exercise #17 Chapter 5 Factoring Let’s work Exercise #19 from Section 5.1

Factor out GCF (3) Terms (2) Terms (4) Terms Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

Factor out GCF (3) Terms (2) Terms (4) Terms Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

Section 5.5 Exercise #20 Chapter 5 Factoring Let’s work Exercise #19 from Section 5.1

Factor out GCF (3) Terms (2) Terms (4) Terms Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

Section 5.6 Quadratic Equations

OBJECTIVES A Solve quadratic equations by factoring.

DEFINITION Quadratic Equation in Standard Form

PROCEDURE Solving Quadratics by Factoring Perform necessary operations on both sides so that right side = 0.

PROCEDURE Solving Quadratics by Factoring Use general factoring strategy to factor the left side if necessary.

PROCEDURE Solving Quadratics by Factoring Use the principle of zero product and make each factor on the left equal 0.

PROCEDURE Solving Quadratics by Factoring Solve each of the resulting equations.

PROCEDURE Solving Quadratics by Factoring Check results by substituting solutions obtained in step 4 in original equation.

Section 5.6 Exercise #24 Chapter 5 Factoring Let’s work Exercise #19 from Section 5.1

Solve. or

Solve. or

Section 5.7 Quadratic Equations

OBJECTIVES A Integer problems. B Area and perimeter problems.

OBJECTIVES Problems involving the Pythagorean Theorem. D Motion problems.

NOTE Terminology Notation Examples: 3,4; – 6,–5 2 consecutive integers n, n+1 Examples: 3,4; – 6,–5

NOTE Terminology Notation 3 consecutive integers n, n+1, n+2 Examples: 7, 8, 9; – 4,– 3,– 2

NOTE Terminology Notation Examples: 8,10; – 6,– 4 2 consecutive even integers n, n +2 Examples: 8,10; – 6,– 4

NOTE Terminology Notation Examples: 13,15; – 21,– 19 2 consecutive odd integers n, n +2 Examples: 13,15; – 21,– 19

DEFINITION Pythagorean Theorem If the longest side of a right triangle is of length c and the other two sides are of length a and b, then

DEFINITION Pythagorean Theorem Hypotenuse c Leg a Leg b

Section 5.7 Exercise #26 Chapter 5 Factoring Let’s work Exercise #19 from Section 5.1

The product of two consecutive odd integers is 13 more than 10 times the larger of the two integers. Find the integers.

The product of two consecutive odd integers is 13 more than 10 times the larger of the two integers. Find the integers.

The product of two consecutive odd integers is 13 more than 10 times the larger of the two integers. Find the integers.

Section 5.7 Exercise #29 Chapter 5 Factoring Let’s work Exercise #19 from Section 5.1

A rectangular 10-inch television screen (measured diagonally) is 2 inches wider than it is high. What are the dimensions of the screen? 10