Factoring by grouping is the best method to use when some terms in a polynomial share one common factor and some other terms in the same polynomial share another common factor. The common factors can then be factored out using the distributive property.

The first step is to determine if all the terms in a polynomial have a common factor. If they do, then the greatest common factor (GCF) can be factored. If they do not, determine if there are two terms in the polynomial that have one common factor and two other terms that have another common factor. Suppose the four-term polynomial is ax + bx + ay + by. Using the commutative property, it can be rearranged, so that ax and ay have the common factor a, and bx and by have the common factor b.

Figure 1: Rearranging terms in a polynomial using the commutative property.

The first two terms, ax and ay, can be factored as a(x + y). The second two terms of the polynomial, bx and by, can be factored as b(x + y). The new polynomial is a(x + y) + b(x + y). Suppose the four-term polynomial is 5m + 2w + mw + 10. It can be rearranged so that 5m + 10 + mw + 2w, or 5(m + 2) + w (m + 2).

Figure 2: Factoring grouped terms

Suppose the polynomial were xy + 3x – 2y – 6. The first two terms can be factored as x(y + 3), and the second two terms can be -2(y + 3). When there is a negative term, the sign must be changed when it is multiplied, so that -2 times a positive 3 is -6. Similarly, suppose the polynomial were x^{2} – 3x + x -3. The first group, x^{2} – 3x, could be factored as x(x – 3). The second group, x -3, actually has one factor in common. The identity factor, 1, can be used, so that the second group can be factored as 1(x – 3).

Figure 3: When factoring negative terms, remember that signs change when multiplying.

The polynomial in symbol form a(x + y) + b(x + y) can be factored using the distributive property as (a + b)(x +y). Similarly, the polynomial 5(m + 2) + w (m + 2) can be factored using the distributive property as (5 + w)(m + 2). The polynomial x (x -3 ) +1(x – 3) can be factored as (x + 1)(x – 3). The identity factor was used to illustrate the distributive property and factoring by grouping.

Figure 4: Use the distributive property after factoring by grouping.

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]]>Factoring a monomial from a polynomial is a process of finding the greatest common factor for the constants, the greatest common factor for the variable terms, and then using the distributive property to factor out the greatest common factor (GCF).

In symbol form, if a ∙ b = c, then both a and b are factors of c. Suppose the monomial is 10x^{4}. There are the factors of 10 and the factors of x^{4}. The factors of 10 are 1, 10, 5, and 2. The factors of x^{4} are x, x^{2}, x^{3}, and x^{4}. They can be combined as 10 ∙ x^{4}, 10x∙x^{3}, 10x^{2}∙ x^{2}, and so on.

Finding the greatest common factor of two constants is a matter of finding the factors of both constants and checking to see what factors are in common. For example, the number 30 can be factored as 2∙3∙5 and the number 42 can be factored as 2∙3∙7. Their GCF, or greatest common factor is 2∙3, or 6.

Figure 1: Finding the GCF of constants.

Finding the GCF of two or more variable terms is also a matter of finding the factors of the variable terms, and seeing what factors are in common. Suppose the terms are x^{5}y^{3}, x^{2}y^{2}, and x^{3}y^{2}. The GCF will be the highest power of the common variables, or x^{2}y^{2}. If the terms were xy^{3}, x^{4}, and xy^{3}, the GCF would be x. All three terms have x in common, because there is no y term in x^{4}, but the highest power is x^{1} or x.

Figure 2: Finding the GCF of terms.

The first step in the process of factoring a monomial from a polynomial is to find the GCF of the constants and the terms. Suppose the polynomial is 3x^{2} + 6x + 9. The GCF of the constants is 3, because 3∙1 is 3, 3∙2 is 6, and 3∙3 is 9. There is no x term in common, because 9 does not have a variable associated with it. The second step in the process is to use the distributive property to factor out the GCF from each of the terms of the polynomial, so that the result is 3(x^{2} + 2x + 3). For example, if the polynomial is 9x^{3} + 27x^{2} + 24x, the constants factor as 3∙3, 3∙9, and 3∙8, so the GCF is 3. There is also a common variable in each term of the polynomial. Therefore, the GCF is 3x, and the result is 3x (3x^{2} + 9x + 8).

Figure 3: Factoring a monomial from a polynomial.

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]]>Solving application problems is a process that includes understanding the problem, translating it into an equation, solving the equation, checking the answer, and answering the question. This process can be used to solve many different types of problems.

The first part of the process involves understanding what is being asked. This includes noticing any key words that refer to operations and any quantities that are in relationship to one another. It is important to have an idea of what sort of quantity will represent a solution. Suppose a problem asks how many points the first-place winner had. That would require just one answer. If the problem described that the first-place winner had 10 points more than the second-place winner, and 17 points more than the third-place winner, all three point values would be necessary to completely answer the question.

Figure 1: Setting up the application problem involves finding quantities in relationship to one another.

The second part of the problem involves putting the problem in symbol form. This includes choosing a letter to represent a variable, and writing down exactly what the variable stands for. Suppose that the point values of the three winners totaled 114. Let the point value of the first-place winner be x, the second-place winner x – 10, and the third-place winner x – 17. Then, x + x – 10 + x – 17 = 114.

Figure 2: Translating the problem into an equation involves putting the problem into symbol form.

If x + x – 10 + x – 17 = 114, then 3x = 114 + 27, or 3x = 141. So, x = 47, x – 10 = 37, and x -17 = 30. To check, 47 + 37 + 30 = 114. The answer makes sense, and fits the parameters of the problem. In this case, the original equation was x + x -10 + x – 17 = 114.

If the work is done to understand the problem before it is set up and solved, it is easier to answer the appropriate question. In this case, all three point values were necessary to completely answer the question. Suppose that more information were added to the problem. During the same contest last year, the point values of the first-place, second-place and third-place winners totaled 100, but the second place winner had 14 points less than the first-place winner, and the third-place winner had 18 points less than the first place winner. Which year did the third-place winner earn more points, and what was the difference? Last year, the equation was x + x – 14 + x – 18 = 100, so 3x = 132. The first-place winner earned 44 points, the second-place winner earned 44-14 or 30 points, and the third-place winner earned 28 points. However, the question asked has 2 parts. Last year, the third-place winner earned 28 points, and this year, the third-place winner earned 30 points. This year, the third-place winner earned more points, and the difference was a gain of 2 points.

Figure 3: Answering the question asked brings all the parts of the problem into balance.

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]]>The first step in solving real-world problems mathematically is to change the problem into an equation. Sometimes, it is a matter of deciding what operations need to be used. It is also a matter of seeing how quantities relate to one another.

Words and phrases that represent addition include such things as added to, increased from, more than, or the sum of. For example, 9 added to a number could be represented algebraically as 9 + x. Words and phrases that represent subtraction include, subtracted from, less than, decreased by, or the difference between. Statements such as multiplied by, the product of, twice or three times, or a certain percent indicate multiplication, and statements such as divided by or the quotient of can indicate division. Sometimes a statement can indicate more than one operation. For example, a statement such as 5 less than 3 times a number could be represented as 3x – 5.

Figure 1: Some words that suggest mathematical operations.

It is also important to determine the relationship between quantities. Suppose that the length of a rectangle is 5 inches longer than its width. The width of the rectangle could be represented as w, so the length is w + 5. When one quantity is given in terms of another, it is usually easier to let one variable stand for the base amount. Any letter other than x can be used, as long as what the variable stands for is clearly defined.

Suppose that one notebook costs $3.00. Then 5 notebooks would cost 5 ∙ 3 or $15.00. If one textbook costs x, then 5 copies of the same text would cost 5x. Similarly, the cost of x items at y dollars can be represented as xy. Sometimes a statement can include more than one operation. Suppose the cost to rent a truck included a daily fee of $40.00, as well as a mileage fee of 60 cents a mile. If the truck were being rented for 2 days, the total cost of the truck would be the daily fee of 40.00 ∙ 2 for the number of days, plus the mileage fee of 0.60x.

Figure 2: Using multiplication and addition to solve a problem.

The word is often means is equal to. Suppose in the truck rental example, the total cost of the truck rental is $260.00. Then $80.00 + 0.60x = 260.00. The equation could then be solved as .60x = 260 – 80, or .60x = 180, or x = 180/.60 = 300 miles.

Figure 3: An example of translating an application to an equation.

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]]>Real numbers include rational and irrational numbers, whole numbers, positive and negative integers, and the number 0. They can be used in operations such as addition, subtraction, multiplication, and division, as well as in expressions involving variables.

Real numbers include many different types of numbers that are familiar, such as positive and negative integers, along the number line. Rational numbers, such as 3/5, are also included, as well as irrational numbers such as π, √7, and √2. The representation of infinity ∞ is not a real number, nor are imaginary numbers.

Figure 1: Contents of the set of real numbers.

Since real numbers fall on the number line, they can be added using the number line model. For example, to add 3 + 2, go from 0 to 3 on the number line, and then forward 2 spaces to 5. Similarly, adding -2 + -5, go left on the number line 2 spaces to -2 and then go left 5 more, to -7. Fractions can be added by finding the least common denominator, then adding them using the number line, because fractions are rational numbers.

Figure 2: Addition of real numbers using the number line model.

Subtraction is the same thing as adding the additive inverse of the number to be subtracted. In symbol form, for any two numbers a and b, a – b is the same thing as a + (-b). Recall that the additive inverse of a number is the opposite of that number, so that a + (-a) = 0, and b + (-b) = 0. Subtraction can also be demonstrated using the real number line model, so that subtracting 7-6 is a matter of starting at 0 and going 7 spaces to 7, then going to the left 6 spaces, to 1.

Figure 3: Definition of subtraction in symbol form.

There are two important rules to remember when multiplying numbers. The product of two positive numbers or two negative numbers is a positive number. The product of a positive number and a negative number is a negative number. For example, 6∙ 2 is 12, as the product of two positive numbers. Similarly -5 ∙ -6 is 30, as the product of two negative numbers. However, -7 ∙ 3 is -21, as the product of a negative number and a positive number, and 4 ∙ -7 is -28, as the product of a positive number and a negative number. The rules for dividing numbers are similar. If both numbers are positive, the quotient will be positive, or if both numbers are negative, the quotient is positive.

Figure 4: Rules for multiplication and division of real numbers.

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]]>Decimals and percent are used in many different types of mathematical calculations and applications. They require attention to detail to use correctly.

In order to add and subtract numbers that contain decimal points, the numbers must be aligned so that the place values are the same. Then they can be added or subtracted as if they were whole numbers. Suppose 3.4 + 90.030 + 5.14 were to be added. In order to line up the numbers around the decimal point, 3.4 could also be written as 3.400, and 5.14 could be written as 5.140. Then 3.400 + 90.030 + 5.140 are all lined up, and equal 98.570. Subtraction is similar. 34.07 -1.569 could be rewritten as 34.070 to line up the decimal points, and then subtract 1.569 to equal 32.501.

Figure 1: Line up decimal points for addition and subtraction.

When multiplying numbers containing decimal points, factors can be multiplied as if they were whole numbers. Then, the number of decimal places in the answer is determined by the number of decimal places to the right in both factors. Suppose that 2.99 is multiplied by 4.8. The answer, 14.352, has three decimal places to the right, two from the factor 2.99 and one from the factor 4.8. Division is the reverse process, with an additional step. Suppose that 12.6 is divided by 2.24. In order to make 2.24 a whole number, both the numerator and the denominator must be multiplied by 100. Therefore, the new ratio is (12.6 ∙ 100)/(2.24 ∙ 100) = 1260/224 = 5.63. Multiplying the numerator and the denominator by 100/100 is the same thing as multiplying by 1, the identity element for multiplication.

When rounding decimals, the place value is always specified. Suppose 36.266 is to be rounded to the nearest hundredth. If the digit to the right of the hundredth place, the thousandth place, is more than 5, then the number in the hundredth place will be rounded up to the next number, or 36.27. Suppose the number 36.313 is to be rounded to the nearest hundredth. The digit 3 in the thousandth place is less than 5. The number remains as 36.31, and the digit in the thousandth place drops off, or the number is truncated.

The word **percent** means “per hundred”. Suppose that the decimal is 0.7. For the percent, the decimal is moved two spaces to the right, or multiplied by 100. If necessary, zeroes are added, so that .7 is 70%. Similarly, .35 is 35%, and 0.012 is 1.2 %.

Figure 3: For decimal to percent, move the decimal two spaces to the right.

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]]>Cube and fourth roots follow rules similar to those for square roots. The rules for rational exponents are an extension of the rules for exponents and roots.

If any real number b^{3} equals c, then b is the cube root of c, when b is not equal to 0. For example, 2^{3} equals 8, so 2 is the cube root of 8. Similarly, 3^{3} equals 27, so 3 is the cube root of 27. The cube root of a positive number is a positive number, because a positive times a positive times a positive is a positive. A negative number, such as -125, can have a real cube root, such as -5, because -5^{3} is -125. Although a negative number times a negative number is a positive number, that positive number times a negative number is a negative number.

Figure 1: The definition of cube roots in symbol form.

If any real number b^{4} equals c, then b is the fourth root of c, when c is a positive number. For example, 4^{4} is equal to 256, so 4 is the fourth root of 256. Similarly, 10^{4} equals 10,000, so 10 is the fourth root of 10,000. The fourth root of a positive number is a positive number. The fourth root of a negative number is not a real number, similar to the square root of a negative number.

A rational exponent such as x^{1/2} is the same thing as the square root of x, √x. Similarly, a rational exponent such as x^{1/3} is the same thing as the cube root of x, and x^{1/4}, as the fourth root of x. The definition can be extended to other rational exponents as well. For example, 8^{2/3} means the same thing as the cube root of 8, which is 2, squared which is 4. Recall that the cube root of 8 is 2, and that 2^{1} is equal to 2. Therefore, x^{1/3} is the cube root of x^{1}, and x^{1} equals x for all real numbers.

Figure 2: Rational exponents of real numbers.

Recall that a negative exponent, a^{-x}, means the same thing as 1/(a^{x}), so that 3^{-2} means the same thing as 1/3^{2}, or 1/9, and so on. That definition holds true for negative rational exponents, so that 27^{-2/3}, means the same thing as 1 over the cube root of 27, which is 3, squared. Then, (1/3)^{2} is 1/9.

Figure 3: Solving a negative rational exponent.

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]]>Linear inequalities in one variable can be solved similarly to linear equations in one variable. Properties that hold for linear inequalities include properties of addition, subtraction, multiplication, and division, as well as those involving relationships between negative numbers.

There are four properties of inequalities that hold true for real numbers. The addition property of equality in symbol form is: if a > b then a + c > b + c. Similarly, for subtraction, if a > b, then a – c > b –c. For the properties of multiplication and division, c must be greater than 0. If a > b and c > 0, then ac > bc. Similarly, if a >b and c > 0, then a/c > b /c. Suppose that the inequality is x -5 >-2. The variable can be isolated on the left side of the equation to solve it, so that x > -2 + 5, or x > 3. Also, a > b means the same thing as b < a.

Figure 1: Properties of inequality in symbol form.

The properties of inequalities for multiplication and division specify that the real number c is greater than 0 for the relationship to hold. What happens if c is a negative number? Suppose the inequality is 5 > 3, and both sides were multiplied by -4. 5(-4) is -20 and 3(-4) is -12. The relationship is -20 < -12, so the direction of the sign changes. Similarly, suppose that 100 > 80, and both sides were divided by -10. 100/-10 is -10 and 80/-10 = -8. The relationship is -10 < -8, so the direction of the sign also changes for division.

Inequalities with the variable on both sides of the sign are solved similarly to equations with the variable on both sides of the equals sign. Suppose the expression is -5q + 9 < -2q + 6. To solve the inequality, -5q + 2q + 9 < -2q + 2q + 6, or -3q + 9 < +6, or -3q < 6 – 9, or -3q < -3. Since dividing by -3 will change the sign, q > 1.

Figure 2: Process of solving inequalities when the variable is on both sides of the sign.

Similar to equations with all real numbers as the solution set, some inequalities have all real numbers as their solution set. Suppose the inequality is 2(x + 3) ≤ 5x -3x + 8. It can be simplified as 2x + 6 ≤ 2x + 8, or 6 ≤ 8. The statement is true for all real numbers.. Suppose the inequality is 3(x + 1) > 3x + 5. Then 3x + 3 > 3x + 5. There is no real number such that 3 > 5, so there is no solution.

Figure 3: An example of an inequality that is true for all real numbers.

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]]>Linear equations can be solved by isolating the variable on one side of the equals sign. This can be done by using the addition and multiplication properties, whether the variable is on one side of the equals sign or there is a variable on both sides of the equals sign.

The goal when solving an equation is to isolate the variable on one side of the equation. That way, the constants are on the other side of the equals sign and the variable is on the other. The variable can be isolated by using the properties of addition, multiplication, or the distributive property of real numbers. Suppose the equation to be solved is 2x + 4 = 20. The variable can be isolated by using the property of addition on both sides of the equation, as 2x + 4 – 4 = 20 – 4. Then 2x = 16. Both sides of the equation can then be divided by 2 so (2x)/2 = 16/2, or x = 8. The solution can be checked by the original equation: 2∙8 + 4 = 20, which is a true statement.

Figure 1: Steps for solving equations.

Like terms or parentheses should be eliminated before isolating the variable on one side of the equation. Suppose the equation were 42x + 3 + 2x + 5 – 8 = 88. The first step would be to combine the like terms using the commutative property, so that 42x + 2x + 3 + 5 – 8 = 88, or 44x + 0 = 88. The variable is easy to isolate as 0 is the identity for addition and can be eliminated, so that 44x = 88. Dividing both sides by 2, using the multiplication property, (44x)/44 = 88/44, or x = 2. Checking the solution, 84 + 3 + 4 + 5 -8 = 88. Suppose the equation were 3y – (2y + 5) = 7. The parentheses can be eliminated first as 3y -2y -5 = 7. Combining like terms, y – 5 = 7 or y = 12.

Solving linear equations when the variable on both sides of the equals sign is similar to solving a linear equation when the variable is on one side of the equals sign. The goal is still to isolate the variable on one side of the equation. Suppose the equation is 5x + 11 = 2x + 20. The variables can be isolated on one side or the other. If the variables are to be located on the left side, 5x – 2x + 11 = 2x – 2x + 20. Combining the like terms in variables, 3x + 11 = 20. Then, 3x = 20 – 11, or 3x = 9. Dividing both sides by 3, (3x)/3 = 9/3, or x = 3. If the variable is to be isolated on the right side, 5x -5x + 11 = 2x -5x + 20. Then, 11 = -3x + 20, 11 – 20 = -3x or -9 =-3x, (-3x)/3 = -9. Since a negative times a positive is a negative, x = 3.

Figure 2: Solving equations with the variable on both sides of the equals sign.

Some linear equations with the variables on both sides have more than one solution. Suppose the equation to be solved is 6x – 3 = 2(3x – 1). Eliminating the parentheses, 6x -3 = 6x -3. Since the equations on both sides of the equals sign are the same equation, any solution for x will be a correct solution. If x = 0, -3 = -3. If x = 1, then 3 = 3. If x = 2, then 9 = 9, and so on. The solution for x is all real numbers. In contrast, some linear equations with the variable on both sides have no solutions. Suppose the equation were 4x + 9 = 2(2x + 5). Eliminating the parentheses, 4x + 9 = 4x + 10. Isolating the variable on one side, 4x -4x + 9 = 4x – 4x + 10 or 9 = 10. That is a nonsense statement, and means that the problem has no solution.

Figure 3: An equation with no solution.

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Real numbers are numbers on the number line. This includes positive and negative numbers, integers, rational numbers, irrational numbers, and the number zero. Real numbers have certain mathematical properties, including commutative, associative, distributive, identity, and inverse properties.

The commutative properties hold for addition and multiplication of real numbers. In symbol form, the Commutative Property for Addition is stated as a + b = b + a. For example, 2 + 4 = 4 + 2. Also, 3 + -4 = -4 + 3. The Commutative Property for Multiplication in symbol form is a∙b = b∙a. For example, 5 ∙ 7 is the same thing as 7 ∙ 5. The commutative property does not hold for subtraction or division. In symbol form, a-b is not the same thing as b-a. For example, 5-3 = 2, but 3-5 = -2, not the same. Also, a/b is not the same thing as b/a. For example, 5/20 equals 1/4, but 20/5 equals 4.

Figure 1: The Commutative Properties for Addition and Multiplication.

The Associative Property of Addition states that for any real numbers a, b, and c, (a + b) + c = a + (b + c). For example, (5 + 6) + 7 equals 11 + 7 or 18. 5 + (6 +7) equals 5 + 13, which also equals 18. The groups of numbers have just changed. The Associative Property of Multiplication states that for any real numbers a, b, and c, (a ∙ b) ∙ c = a ∙ (b ∙ c). For example, (3∙ 2) ∙ 5 equals 6∙ 5, or 30, and 3 ∙ (2∙5) equals 3∙ 10, or 30.

Figure 2: The Associative Properties for Addition and Multiplication.

The Distributive Property involves 2 operations, multiplication and addition. In symbol form, a (b + c) = a∙b + a∙ c. For example, 5(2 + 2) = 20, and so does 10 + 10. There is also a Distributive Property of Multiplication over Subtraction such that a ∙ (b-c) = a∙ b –a ∙c. For example, 7 ∙ (5 -2) = 21, and so does 35 – 14.

Figure 3: The Distributive Property distributes multiplication over addition (or subtraction).

The identity element for addition is 0. Any real number plus 0 equals the number. The identity element for multiplication is 1. Any real number times 1 equals the number. The additive inverse for any real number a is –a, and a + -a = 0, the identity element for addition. The multiplicative inverse for a, a not equal to 0, is its reciprocal, 1/a. In symbol form, a ∙ 1/a = 1, the identity element for multiplication.

Figure 4: Adding a real number and its additive inverse = 0, the additive identity, and multiplying a real number times its multiplicative inverse = 1, the multiplicative identity.

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