study tool for the students who search educational notes on internet, Schemes and Mind Maps of Mathematics

Download study tool for the students who search educational notes on internet and more Schemes and Mind Maps Mathematics in PDF only on Docsity! Algebra For Beginners - Basic Introduction Let's start with an example: 3x + 5 + 4x - 2. To add these terms, we first need to identify the like terms. In this case, 3x and 4x are like terms. We add the coefficients (3 + 4 = 7) and keep the x variable. So, the answer is 7x. Next, we add the constants: 5 - 2 = 3. Therefore, the final answer is 7x + 3. Now, let's move on to adding two trinomials. For example: 4x^2 + 3x + 9 + 5x^2 + 7x - 4. We start by adding the like terms: 4x^2 and 5x^2. 4 + 5 = 9, so the answer is 9x^2. Next, we add 3x and 7x: 3 + 7 = 10, so the answer is 10x. Finally, we add 9 and -4: 9 - 4 = 5. Therefore, the final answer is 9x^2 + 10x + 5. Subtracting Like Terms Now, let's move on to subtracting two trinomials. For example: 5x^2 - 6x - 12 - 7x^2 + 4x - 13. We start by distributing the negative sign to all terms on the right side of the equation. This changes the signs of the terms. Next, we combine like terms: -6x and 4x. -6 + 4 = -2, so the answer is -2x. Finally, we combine -12 and -13: -12 + 13 = 1. Therefore, the final answer is -2x^2 - 10x + 1. Multiplying Monomials Multiplying monomials involves multiplying the coefficients and adding the exponents of the variables. For example: x^3 * x^4. When we multiply these monomials with the same base (x), we add the exponents (3 + 4 = 7). So, the answer is x^7. Let's try another example: x^5 * x^7. Adding the exponents (5 + 7) gives us x^12 as the answer. Dividing Monomials Dividing monomials involves dividing the coefficients and subtracting the exponents of the variables. For example: x^8 / x^3. When we divide these monomials with the same base (x), we subtract the exponents (8 - 3 = 5). So, the answer is x^5. Let's try another example: x^5 / x^2. Subtracting the exponents (5 - 2) gives us x^3 as the answer. Remember, x is just a variable that represents an unknown value. It can be any number. Multiplying Monomials with Multiple Variables When multiplying monomials with multiple variables, we multiply the coefficients and add the exponents of each variable. For example: 4xy^2 * 8x^2y^3. Multiplying the coefficients (4 * 8) gives us 32. Next, we multiply the x variables: x * x^2 = x^3. Finally, we multiply the y variables: y^2 * y^3 = y^5. Therefore, the answer is 32x^3y^5. Dividing Monomials with Multiple Variables When dividing monomials with multiple variables, we divide the coefficients and subtract the exponents of each variable. For example: 12x^9y^5 / 4x^3y^12. Dividing the coefficients (12 / 4) gives us 3. Next, we subtract the exponents of the x variables: 9 - 3 = 6. Finally, we subtract the exponents of the y variables: 5 - 12 = -7. Since we have a negative exponent for y, we move it to the denominator and change the sign. Therefore, the answer is 3x^6 / y^7. Practice Problems Now, let's work on some practice problems: 1. Multiply: 5x^2y^3 * 6x^3y^4 2. Multiply: 7a^3b^4 * 8a^5b^7 3. Divide: 12 / 36x^3y^7 4. Divide: 56a^8b^11 / a^3b^4 Remember to simplify the coefficients and the exponents of the variables. When factoring expressions, it is important to find the greatest common factor (GCF) of the terms. The GCF is the largest factor that divides evenly into all the terms. Let's look at some examples: Example 1: Factor the expression: 4x^2 + 2x The GCF of 4 and 2 is 2. Since both terms contain at least one x variable, the GCF will also contain x. Dividing each term by the GCF, we get: 4x^2 ÷ 2x = 2x 2x ÷ 2 = 1 Therefore, the factored form of the expression is 2x(x + 1). Example 2: Factor the expression: 12ab^2 + 18a^2b^3 The GCF of 12 and 18 is 6. The GCF of a and a^2 is a. The GCF of b^2 and b^3 is b^2. Dividing each term by the GCF, we get: 12ab^2 ÷ 6ab^2 = 2 18a^2b^3 ÷ 6ab^2 = 3ab Therefore, the factored form of the expression is 6ab(2 + 3ab). Example 3: Factor the expression: x^2 - 25 This is a difference of perfect squares. The formula to factor a difference of squares is: a^2 - b^2 = (a + b)(a - b). In this case, a is x and b is 5. Therefore, the factored form of the expression is (x + 5)(x - 5). Example 4: Factor the expression: x^2 - 9 This is also a difference of perfect squares. Using the formula, we get: (x + 3)(x - 3). Example 5: Factor the expression: 4x^2 - 25 This is a difference of perfect squares. Using the formula, we get: (2x + 5)(2x - 5). Example 6: Factor the expression: 25x^2 - 16y^2 This is a difference of squares. Using the formula, we get: (5x + 4y)(5x - 4y). Example 7: Factor the expression: 81x^4 - 16y^8 This is also a difference of squares. Using the formula, we get: (9x^2 + 4y^4)(9x^2 - 4y^4). However, we can further factor the second part as a difference of squares: (9x^2 + 4y^4)(3x + 2y^2)(3x - 2y^2). Factor by Grouping: When factoring polynomials with four terms, we can use a technique called factor by grouping. The key is to look for a common factor in the first two terms and the last two terms: Example 8: Factor the polynomial: 2x^3 - 6x^2 + 4x - 12 In this case, the ratio between the coefficients of the first two terms (-4/1) is the same as the ratio between the coefficients of the last two terms (-12/3). This indicates that we can use factor by grouping. Factoring out the GCF in the first two terms, we get: 2x^2(x - 3) Factoring out the GCF in the last two terms, we get: 4(x - 3) Now we have a common factor of (x - 3), so our final factored form is: (x - 3)(2x^2 + 4). Example 9: Factor the polynomial: 3x^3 + 8x^2 - 6x - 16 In this case, the ratio between the coefficients of the first two terms (3/1) is the same as the ratio between the coefficients of the last two terms (-6/-2). This indicates that we can use factor by grouping. Factoring out the GCF in the first two terms, we get: x^2(3x + 8) Factoring out the GCF in the last two terms, we get: -2(3x + 8) Now we have a common factor of (3x + 8), so our final factored form is: (3x + 8)(x^2 - 2). These are some of the techniques used to factor expressions and polynomials. Practice these methods with different examples to become more comfortable with factoring. Study Hack: Understanding Multiplication of Polynomials In algebra, it's important to understand how to multiply different types of polynomials. In this article, we'll explore the multiplication of monomials, binomials, trinomials, and more. We'll also discuss factoring, which is the reverse process of multiplication. Multiplying Monomials When multiplying monomials, the exponent rules come into play. For example, any number raised to the power of zero is one. So, even if a variable is raised to the zero power, the answer is one.  Example: 50 = 1  Example: 30 = 1 Now, let's consider the difference between the following three problems: 1. -23 2. (-2)3 3. -23 In the first example, we have a single negative sign, so the answer is -8. In the second example, the negative sign is inside  4x^2 - 25 can be factored as (2x + 5)(2x - 5).  16x^2 - 81 can be factored as (4x + 9)(4x - 9).  25x^2 - 16y^2 can be factored as (5x + 4y)(5x - 4y).  81x^4 - 16y^8 can be factored as (9x^2 + 4y^4)(9x^2 - 4y^4). Lastly, if we have a polynomial with four terms, we can use the technique of factor by grouping. We group the terms and factor out the GCF from each group. For example, x^3 - 4x^2 + 3x - 12 can be factored as (x^2 - 4)(x + 3). When factoring by grouping, you can take out the greatest common factor (GCF) in each pair of terms. Let's look at an example: First, consider the expression x2 + 3. The GCF here is 1, so we can write it as 1(x2 + 3). Now, let's focus on the x2 term. If we divide it by x - 4, the x - 4 terms will cancel out, leaving us with just x2. Similarly, if we divide the 3 term by x - 4, the x - 4 terms will cancel out, leaving us with just 3. So, the factored form of the expression is x2 + 3. Let's try another example: 2x3  - 6x2 + 4x - 12. First, let's find the GCF in the first two terms, which is 2x2. Dividing 2x3 by 2x2 gives us x, and dividing -6x2 by 2x2 gives us -3. Now, let's find the GCF in the last two terms, which is 4. Dividing 4x by 4 gives us x, and dividing -12 by 4 gives us -3. Notice that the two -3 terms are the same, which means we're on the right track. So, the factored form of the expression is (x - 3)(2x2 + 4). To confirm the factored form, you can expand it using the FOIL method. Let's do that: x(x - 3) gives us x2  - 3x, and 3(x - 3) gives us 3x - 9. Combining these terms, we get x2  - 3x + 3x - 9, which simplifies to x2  - 9. This matches the original expression, so our factored form is correct. Let's try one more example: 3x3 + 8x2  - 6x - 16. The GCF in the first two terms is x2. Dividing 3x3 by x2 gives us 3x, and dividing 8x2 by x2 gives us 8. The GCF in the last two terms is - 2. Dividing -6x by -2 gives us 3x, and dividing -16 by -2 gives us 8. So, the factored form of the expression is (3x + 8)(x2  - 2).