Grade 8 Pre-Algebra Introduction to Algebra. CONFIDENTIAL

Warm Up Solve the equation: 1) 2 = 7 - x 1) x = 5 2) - 3b = 21 CONFIDENTIAL

Here is a typical polynomial: Introduction Constants and variable: A symbol in algebra having a fixed value is called a constant, whereas a symbol which can be assigned different values is called a variable. Example: 1/3, -8, Π,√2 are all constants and x, y, z are all variables. Algebraic expression: A combination of constants and variables connected by signs +, -, × and ÷ is called an algebraic expression. The several parts of an expression separated by + or - sign are called the terms of an expression. Terms Here is a typical polynomial: 9x2 + 2x - 5 CONFIDENTIAL

Example: 3x3 is a monomial. Monomial: is a variable that is formed with a number and a letter variable to its powers. Example: 3x3 is a monomial. You can’t add or subtract monomials if they have different exponents such as 3x3 and 4x4. But you can multiply or divide them. To multiply monomials, just add the exponents of the variables and multiply the coefficients. 3x3 x 4x4 = 12x7. Here are some additional ways to manipulate the monomials: (am)n = amn (ab)m= ambm CONFIDENTIAL

Polynomial: is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Example: 7x4 + 6x2 + x is a polynomial . Each piece of the 7x4 + 6x2 + x , each part that is being added, is called a "term". When a term contains both a number and a variable part, the number part is called the "coefficient". The coefficient on the leading term is called the leading coefficient. coefficient 9x2 + 2x - 5 Leading coefficient is 9. CONFIDENTIAL

Here are some examples: 6x –2 NOT a polynomial term. Polynomial terms have variables to whole-number exponents; there are no square roots of exponents, no fractional powers, and no variables in the denominator. Here are some examples: 6x –2 NOT a polynomial term. This has a negative exponent. 1 This has the variable in the denominator. sqrt(x) This has the variable inside a radical. 4x2 A polynomial term . x2 CONFIDENTIAL

7x4 + 6x2 + x is a polynomial in x of degree 4. Degree: If the polynomial is of one variable, then the highest power of the variable is called the degree if the polynomial. 7x4 + 6x2 + x is a polynomial in x of degree 4. Linear polynomial: is a polynomial of degree 1. Example: (4x + 2), (3/5 + 7x) is a linear polynomial . An equation says that two things are equal. It will have an equals sign "=" like this: Example: 4a + 5 =13 is an equation. CONFIDENTIAL

ADDITION AND SUBTRACTION OF POLYNOMIALS: Collect the like terms together. Find the sum or difference of the numerical coefficients of these terms. The resulting expression should be in the simplest form and can be written according or descending order of terms. CONFIDENTIAL

Add: (4y2 + y - 6) + (6y2 - 4y + 9) = (4y2 + 6y2) + (y - 4y) - 6 + 9 Subtract: (9y2 + 3) - (5y2 + 6) = (9y2 - 5y2) + 3 - 6 = 4y2 - 3 CONFIDENTIAL

6) The degree of the polynomial 7y4 - 10y is _______. Now you try! Add: 1) 4a + 2x + 6y 1) 6y + 9x and 4a - 7x 2) 8y2 + 2y 2) 4y2 + 5y and 4y2 - 3y -11 5 2 3 3) -5x2y2 + x2y2 , 7x2y2 , x2y2 7 15 3) x2y2 Subtract: 4) -7x2y from -9x2y 4) -9x2y 5) 9y2 + 3 from 5y2 + 6 5) -4y2 + 4 6) 4 6)  The degree of the polynomial 7y4 - 10y is _______. CONFIDENTIAL

9)Which of the following expressions is a monomial? 7y 4 - y Now you try! 7) 7y Solve: 7) Find the width of the rectangular field, if the perimeter is 16y + 8 and the length is y + 4. 8) The two sides of a triangle are 2y and 4y. Perimeter of the triangle is (8y + 1). Find the third side. 8) 2y + 1 9)Which of the following expressions is a monomial? 7y 4 - y y2 + 4y - 7 CONFIDENTIAL

MULTIPLITION OF POLYNOMIALS: Case 1: Multiplication of monomials Product of monomials = Product of numerical coefficient x Product of literal factors Case 2: Multiplication of polynomials Multiply each term of one polynomial with each term of the other polynomial and simplify by taking the like terms together. CONFIDENTIAL

Multiply: 16x2y3 by 4x4y3z = (16 x 4) ×(x2y3 × x2y3z) = 64x6y6z 4x2 - 6x + 5 by 3x + 2 = 3x(4x2 - 6x + 5) + 2(4x2 - 6x + 5) = (12x3 - 18x2 + 15x) + (8x2 - 12x + 10) = (12x3 - 10x2 + 3x + 10) CONFIDENTIAL

Now you try! Multiply: 1) (9y2 + 3) and (5y2 + 6) 2) 4y2 + 5y and 4y2 - 3y 3) 6y + 9x and 4a - 7x 1) 45y4 + 69y2 + 18 2) 16y4 + 8y3 - 15y2 3) 24ay + 36ax - 42xy- 63x2 CONFIDENTIAL

DIVISION OF POLYNOMIALS: Case 1: Division of monomial by a monomial Quotient of two monomials = Quotient of numerical coefficient x Quotient of literal factors Case 2: Multiplication of polynomial by a monomial Divide each term of the polynomial by the monomial. CONFIDENTIAL

Divide: 32x3y3 by -8xy = 32x3y3 = 32 × (x3y3) -8xy -8 ( xy) = -4x2y2 18x4y2 + 15x2y2 - 27x2y by -3xy = 18x4y2 + 15x2y2 - 27x2y -3xy -3xy -3xy -3xy = -6x3y - 5xy + 9x CONFIDENTIAL

Divide: (14x2 -53x + 45) by (7x - 9) 2x - 5 7x - 9 14x2 -53x + 45 CONFIDENTIAL

2) (3ab2c3 - 2a2b2c2 + ab2c) by (abc) 4 5 3 2 Now you try! Divide: 1) -9a3b2c2 1) -81a5b4c3 by -9a2b2c 2) (3ab2c3 - 2a2b2c2 + ab2c) by (abc) 4 5 3 2 3) (x3 -4x2 + 7x - 2) by (x - 2) 2) (3bc2 - 4abc + 4b) 2 5 3 3) (x2 - 2x + 3) CONFIDENTIAL

Algebraic Identities Consider the statement (x + 3)2 = x2 + 6x + 9. x LHS RHS 1 16 2 25 3 36 4 49 On putting various values of x, you find that L.H.S = R.H.S. for all values of x. Such a mathematical sentence containing an unknown variable x which is satisfied for all values of x is called an identity. CONFIDENTIAL

IDENTITY 1: (x + a) (x + b) = x2 + (a + b)x + ab Some identities IDENTITY 1: (x + a) (x + b) = x2 + (a + b)x + ab IDENTITY 2: (x + a)2 = x2 + 2ab + b2 IDENTITY 3: (x - a)2 = x2 - 2ab + b2 IDENTITY 4: (a + b) (a - b) = a2 - b2 Other formulas: 1) a2 + b2 = (a + b)2 - 2ab 2) a2 + b2 = (a - b)2 + 2ab 3) (a - b)2 = (a + b)2 - 4ab 4) (a + b)2 = (a - b)2 + 4ab 5) (a + b)2 + (a - b)2 = 2(a2 + b2 ) CONFIDENTIAL

Here are some examples: 1) (a + 3)2 = a2 + 2×a×3 + 32 = a2 + 6a + 9 2) (b - 5)2 = b2 - 2×b×5 + 52 = b2 - 10b + 25 3) (2x + 3)(2x - 3) = (2x)2 - 32 = 4x2 - 9x 4) Find a2 + b2 when (a + b) = 7 and ab = 12 a2 + b2 = (a + b)2 - 2ab = (7)2 - 2×12 = 49 - 24 = 25 CONFIDENTIAL

5) Find a2 + b2 when (a - b) = 5 and ab = 14 Now you try! Solve: 1) (3a + 7)2 1) 3a2 + 21a + 49 2) (2y + 5)(2y - 5) 2) 4y2 -25 3) (2m - 3n)2 3) 4m2 - 12mn + 9n2 4) 3a + b 2 b 2a 4) 9a2 - 3+ b2 b2 4a2 5) Find a2 + b2 when (a - b) = 5 and ab = 14 5) 53 CONFIDENTIAL

Factorize: 1) a2 + 4a + 4 = (a)2 + 2.2.a + (2)2 = (a + 2)2 2) 9a2 - 4b2 = (3a)2 - (2b)2 = (3a + 2b) (3a - 2b) 3) a2 - b2 = a 2 - b 2 = a + b a - b 25 36 5 6 5 6 5 6 CONFIDENTIAL

Now you try! Factorize: 1) 64a2 + 80ab + 25b2 1) (8a + 5b)2 3) 4a + 2b a - 2b 5b 10a 5b 10a CONFIDENTIAL

Now you try! 1) The volume of a cube is (5x3 + 8) m3. A smaller cube with volume (x3 + 5) m3 is cut out of the cube. Choose a polynomial for the remaining volume.  1) (4x3 + 3) m3 2) Two square playgrounds have areas 4x2 and 9y2. Express the difference in area in a factored form. 2) (2x + 3y) (2x - 3y) 3) Which of the numbers out of 3 and 4 is a solution of the equation, 3x + 2 = 14? 3) 3 CONFIDENTIAL

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Assignments 1) Add : 8y3 + 4y2 - 8y + 5 and - 8 - 4y3 + 5y2 + 7y. 2) Subtract: 7y - 4y2 - 6 from 2y + 3 - y2. 2) 3y2 - 5y + 9 3) Simplify: (2y + 5)(2y - 5) 3) 4y2 - 25 4)Which of the following is an equation? I. 2x - 3 II. 2y + 2 < 4 III. x - 3 = 0 4) III. x - 3 = 0 CONFIDENTIAL

5) The degree of the polynomial 9y8 + 1 is_____. 5) 8 5) The degree of the polynomial 9y8 + 1 is_____. 6) Which of the numbers out of 2 and 3 is a solution of the equation, 6x + 5 = 23? 6) 3 7) Find 3y + 4z, for y = 6 and z = 3. 7) 30 8) Robert and his friends went to a magic show with their families. Cost of ticket for each adult was $25 and for each child was $12. The group had 14 adults and the total cost for the tickets was $518. How many children were there in the group? 8) 12 CONFIDENTIAL

9) Ashley's salary 3 years ago was $y  9) Ashley's salary 3 years ago was $y. Now, she gets 4 times the salary and spends $2,598. Calculate her savings. 9) 3y -2598 10) 3(5x2 + 6)(x + 1)(x - 1) 10) Factorize: 15x4 - 3x2 - 18 11) Factorize: 1 - 121 a2 12) Factorize: 2x2 + 7xy - 15y2 11) 1 + 11 1 - 11 a a 12) (2x - 3y)(x + 5y) CONFIDENTIAL

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