Factoring and Applications

Solving Quadratic Equations by Factoring:

A quadratic equation is an equation of the form ax² + bx + c = 0, where a ≠ 0. This is called the standard form of a quadratic equation. The highest degree of any term in a quadratic equation is 2. To solve a quadratic equation by factoring: Make sure the equation is in standard form. That is, ax² + bx + c = 0 Factor, if possible. Set each factor equal to zero. Solve the resulting equation(s) to find each solution. Check each solution in the original equation. There will be, at most, two solutions to a quadratic equation. Sometimes both solutions will be the same number. This is called a "double root." A double root occurs when the quadratic expression is a perfect square trinomial = (x - a)² = 0 6x² - 12x = 0 = 6x(x - 2) = 0 6x = 0 x - 2 = 0 +2 +2 x = 0 x = 2 Answer = x = 0, 2 x² + 4x = 0 = x(x + 4) = 0 x = 0 x + 4 = 0 - 4 - 4 Answer = x = 0, - 4 2x² - 8x - 42 = 0 = 2(x2 - 4x - 21) = 0 = 2(x - 7)(x + 3) = 0 x - 7 = 0 x + 3 = 0 +7 +7 - 3 - 3 Answer = x = 7, -3 10x² - x = 2 = 10x² - x - 2 = 0 = (5x + 2)(2x - 1) = 0 5x + 2 = 0 2x - 1 = 0 - 2 - 2 + 1 + 1 / 5 / 5 / 2 / 2 Answer = x = -2/5, ½ -5t² + 13t + 6 = 0 = (-1)-5t² + 13t + 6 = 0(-1) to get a positive coefficient, multiply both sides by -1 = 5t² - 13t - 6 = 0 = (5t + 2)(t - 3) 5t + 2 = 0 t - 3 = 0 - 2 - 2 +3 +3 / 5 / 5 Answer = t = -2/5, 3 4x² + 9 = 12x - 12x - 12x = 4x² - 12x + 9 = 0 = (2x - 3)² = 0 = (2x - 3)(2x - 3) = 0 2x - 3 = 0 2x - 3 = 0 +3 +3 +3 +3 / 2 / 2 / 2 / 2 Answer = x = 3/2 x² - 64 = 0 = (x + 8)(x - 8) = 0 x + 8 = 0 x - 8 = 0 - 8 - 8 +8 +8 Answer = x = -8, 8

Introduction to Solving Quadratic Equations:

A quadratic equation is an equation that can be written in the form ax² + bx + c = 0. This is called the standard form of a quadratic equation. The highest degree of any term in a quadratic equation is 2. x² + 6 = -4x = x² + 6 + 4x = -4x + 4x = x² + 4x + 6 = 0 8x + 1 = 5x² = 8x + 1 - 8x - 1 = 5x² - 8x - 1 = 5x² - 8x - 1 = 0 -7x² = 8 = -7x² + 7x² = 8 + 7x² = 7x² + 8 = 0 3x + 10x² = -2 = 10x² + 3x + 2 = -2 + 2 = 10x² + 3x + 2 = 0 7x² = 5x + 8 = 7x² - 5x - 8 = 5x + 8 - 5x - 8 = 7x² - 5x - 8 = 0 6x - 2x² = -3 = 6x - 2x² - 6x + 2x² = -3 - 6x + 2x² = 0 = 2x² - 6x - 3 4x² = 9 = 4x² - 9 = 9 - 9 = 4x² - 9 = 0

Factoring by Grouping

Group the terms together so that each group has a common factor. Factor within each group. The remaining factor in each group should be the same. Factor the entire polynomial. Multiply to check you answer. 6x² - 15x + 4x - 10 Group terms (6x² - 15x) + (4x - 10) = 3x and 2 Factor within groups 3x(2x - 5) + 2(2x - 5) Factor the entire polynomial (2x - 5)(3x + 2) Answer 6x² - 15x + 4x - 10 = (2x - 5)(3x + 2) 2x² + 5x - 4x - 10 Group terms (2x² + 5x) - (4x - 10) Factor within groups x(2x + 5) - 2(2x + 5) Factor the entire polynomial (2x + 5)(x - 2) Answer 2x² + 5x - 4x - 10 = (2x + 5)(x - 2)

Special Cases of Factoring:

Identifying a Difference of Two Squares: a² - b² = (a + b)(a - b), where a and b are numbers or algebraic expressions. An expression is a difference of two squares if 1. the expression id a binomial 2. both terms are perfect squares 3. and, the terms are subtracted 25x² - 16 First Term = 25x² = (5x)² Last Term = 16 = (4)² Sign = - Result = Yes 49r² - 10 First Term = 49r² = (7r)² Last Term = 10 = 10 is not s perfect square Sign = - Result = No 9x² + 64y⁴ First Term = 9x² = (3x)² Last Term = 64y⁴ = (8y2)² Sign = + Result = No m¹² - 1 First Term = m¹² = (m6)² Last Term = 1 = (1)² Sign = - Result = Yes x³ - 36 First Term = x³ is not a perfect square Last Term = 36 = (6)² Sign = - Result = No

Factoring Trinomials of the Form x² + bx + c: RULE

The answer will be in the form of (x + __)(x + __) The two blanks are numbers such that, when you multiply them, you get the last term, which is c; and when you add them, you get the coefficient of the middle term, which is b.

Graphs of Quadratic Equations:

The graph of a quadratic equation y = ax² + bx + c, where y = 0, is called a parabola. A solution to an equation is the number(s), that when substituted for the variable(s), makes the equation true. The x-intercept(s) of the graph of an equation is/are the point(s) where the curve crosses the x-axis. For any x-intercept, the value of y is 0. NOTE: Some graphs may have one or more x-intercepts, while others may not have any. The solution(s) to the quadratic equation ax² + bx + c = 0 are the x-values of the x-intercept(s) of the graph of y = ax² + bx = c.

Factoring Polynomials:

To factor a polynomial: Look for a common factor, if there is one, factor out the GCF. Look at the numbers of terms. Two Terms = Look to see if the polynomial is a difference of two squares. The factored form of a² - b² is (a + b)(a - b) Three Terms = Determine whether the polynomial is a perfect square trinomial. The factored form of a² - 2ab + b² is (a - b)² and a² + 2ab + b² -s (a + b)², if not, try to factor by the reverse FOIL method. Four Terms = Try to factor by grouping. Always factor completely by making sure that all common factors are factored out and that each factor is also prime. Check by multiplying. 6x² - 486 = 6(x² - 81) Both the first and second terms are squared and the terms are being subtracted, therefore this binomial is a difference of two squares = 6(x + 9)(x - 9) 3k² - 48 = 3(k² - 16) = k² - 16 = (k)² - (4)² This binomial is a difference of two squares = 3(k + 4)(k - 4) 12n³ - 12n² - 144n = 12n(n² - 1n - 12) = n² - 1n - 12 = (n + __ )(n - __ ) Factor Pairs = 1, -12 = 1 + -12 = -11 -1, 12 = -1 + 12 = 11 2, -6 = 2 + (-6) = -4 -2, 6 = -2 + 6 = 4 3, -4 = 3 + (-4) = -1 -3, 4 = -3 + 4 = 1 = 12n(n + 3)(n - 4) x⁴ - 10x² + 25 = (x²)² - 2(5x²) + (5)² = (x² - 5)² x³ + 2x² - 9x - 18 = (x³ + 2x²)(-9x - 18) = x²(x + 2) - 9(x + 2) = (x + 2)( x2 - 9) = (x + 2)(x + 3)(x - 3) 4x³ + 8x² = 4x²(x + 2) 12n² - 7n - 7 = PRIME (cannot be factored)

Factoring Using the Reverse FOIL Method:

To factor the trinomial ax2 + bx + c using the reverse FOIL method, list the different factorizations of ax2, and list the different factorizations of c. List the possible factoring combinations until the correct one is reached. If c is positive, then both the signs of the binomial factors are the same as the sign of b. If c is negative, then the signs of the binomial factors are different. Check the middle term of each combination by computing the Outer and Inner products. The one with the correct middle term is the correct factorization. 2x² + 7x + 5 List the factorization of 2x² = (x)(2x) List the different factorizations of 5 = 1 x 5 and 5 x 1 List the possible factoring combinations = (x + 1)(2x + 5) Answer = 2x² + 7x + 5 = (x + 1)(2x + 5)

Factoring a Trinomial:

To factor trinomials in the form x² + bx + c, do the following: 8x² + 8x - 6 Factor out the GCF if there is one. GCF is 2 = 2(4x² + 4x - 3) Factor the resulting trinomial. 4x² + 4x - 3 List the factorization of 4x2 = (x)(4x) List the factorization of -3 = (1)(-3) = (x + 1)(4x - 3) Check the middle term by computing the Outer and Inner products = x = NO List the factorization of 4x² = (x)(4x) List the factorization of -3 = (3)(-1) = (x + 3)(4x - 1) Check the middle term by computing the Outer and Inner products = 11x = NO List the factorization of 4x² = (2x)(2x) List the factorization of -3 = (1)(-3) = (2x + 1)(2x - 3) Check the middle term by computing the Outer and Inner products = -4x = NO List the factorization of 4x² = (2x)(2x) List the factorization of -3 = (-1)(3) = (2x - 1)(2x + 3) Check the middle term by computing the Outer and Inner products = 4x =YES Answer = 2(2x - 1)(2x + 3)

Applications of Quadratic Equations:

When an object is thrown or launched upward, its approximate height in meters is given by the quadratic equation S = -5t² + vt + h, where S is the height in meters, v is the initial velocity, t represents the time in seconds after the object is thrown or launched, and h represents the initial height of the object (that is, the height above the ground of the object when it was first thrown or launched). A ball is thrown upward with an initial velocity (v) of 13 meters per second. Suppose that the initial height (h) above the ground is 6 meters. At what time (t) will the ball hit the ground? The ball is on the ground when S = 0. Use the equation S = -5t² + vt + h. 0 = -5t² + 13t + 6 Multiply both sides of the equation by -1 (-1) 0 = -5t² + 13t + 6 (-1) = 5t² - 13t - 6 = 0 = (5t + 2)(t - 3) = 0 5t + 2 = 0 t - 3 = 0 - 2 - 2 +3 +3 5t = - 2 t = 3 Answer = the ball will hit the ground in 3 seconds

Factoring Trinomials of the Form x² + bx + c: PROCEDURE

Write the first terms of the binomial factors, (x + __)(x + __). Find two numbers whose product is c and whose sum is b. Put these two numbers in the blanks. Multiply the factored answer to check if the product is the original trinomial. x² + 8x + 12 (x + __)(x + __) List the possible pairs of factors of 12. Since 12 is positive, both factors will either be positive or both will be negative. Since 8x is positive, both factors must be positive. 1 x 12, 2 x 6, and 3 x 4 Find two numbers whose product is 12 and whose sum is 8. Factors = 1 and 12 Sum = 13 Factors = 2 and 6 Sum = 8 Factors = 3 and 4 Sum = 7 Write these two numbers as the constants in the binomial factors = (x + 2)(x + 6) Answer = x² + 8x + 12 = (x + 2)(x + 6)

Prime Polynomial

a polynomial that cannot be factored. 3x + 7y + 12xy = GCF = 1 = 3x + 7y + 12xy = PRIME

Factoring a Polynomial by the GCF

find the GCF of the terms in the polynomial. Write each term as the product of the GCF and each terms remaining factors. Use the reverse of the Distributive Property to factor out the GCF. NOTE* there should not be any common factors in the remaining polynomial factor. Check by multiplying. 9x⁵ + 18x² + 3x 9x⁵ = (9)(x)(x)(x)(x)(x) 18x² = (18)(x)(x) 3x = (3)(x) = GCF = 3x 9x⁵ = 3x(3x⁴) 18x² = 3x(6x) 3x = 3x(1) = 3x(3x⁴ + 6x + 1)

Factoring a Polynomial

is the process of writing the polynomial as a product of two or more factors. Factoring a polynomial changes a sum and/or difference of terms into a product of factors. x² = (x)(x) x³ = (x)(x)(x) = GCF = x²

Finding the GCF of Variable Terms

the GCF of two or more terms with the same variable, is found by choosing the variable in the list that is raised to the smallest exponent. x³ = (x)(x)(x) x⁷ = (x)(x)(x)(x)(x)(x)(x) x⁵ = (x)(x)(x)(x)(x) = GCF = x³ y = (y) y⁴ = (y)(y)(y)(y) y⁷ = (y)(y)(y)(y)(y)(y)(y) = GCF = y x = (x) y² = (y)(y) = GCF = 1

Greatest Common Factor (GCF)

the greatest common factor (GCF) of two or more numbers is the largest number that divides exactly into each of the numbers.

Factor

to "factor" means "to write as a product," when two or more numbers, variable, or algebraic expressions are multiplied, each is called a factor. In the product 2 x 3 = 6, the numbers 2 and 3 are factors of 6, and 2 x 3 is the factored form of 6. Factoring is the reverse of multiplying. You can check if you have factored correctly by multiplying and seeing if the original expression results.

Finding the GCF of Terms

to find the GCF of two or more terms with coefficients and variables, find the product of the GCF of the coefficients and the GCF of the variable factors. 9x² = (9)(x)(x) 15x³ = (15)(x)(x)(x) = GCF = 3x²