Algebra Division and Remainder Theorem

Division

Remainder Theorem

• The Remainder Theorem states that when a polynomial f(x) is divided by (x - a), the remainder is f(a).
• This theorem can be used to find the remainder of a polynomial division without actually performing the division.
• It is often used to test if a polynomial has a certain root or factor.

Word Problems

• Division word problems often involve sharing, grouping, or measuring.
• Examples of division word problems:
  • Sharing: "If 12 cookies are shared equally among 4 friends, how many cookies will each friend get?"
  • Grouping: "If 18 pencils are placed in boxes of 3, how many boxes can be filled?"
  • Measuring: "If 24 ounces of juice are poured into 4-ounce cups, how many cups can be filled?"
• To solve division word problems, identify the total amount, the number of groups, and the number of items per group.

Divisibility Rules

• Divisibility rules are used to determine if a number is divisible by another number without performing the actual division.
• Common divisibility rules:
  • Divisible by 2: last digit is even (0, 2, 4, 6, or 8)
  • Divisible by 3: sum of digits is divisible by 3
  • Divisible by 4: last two digits form a number divisible by 4
  • Divisible by 5: last digit is 0 or 5
  • Divisible by 6: divisible by both 2 and 3
  • Divisible by 9: sum of digits is divisible by 9
  • Divisible by 10: last digit is 0
• These rules can be used to quickly determine if a number is divisible by another number.

Division

Remainder Theorem

• States that when a polynomial f(x) is divided by (x - a), the remainder is f(a)
• Allows finding the remainder of a polynomial division without performing the actual division
• Often used to test if a polynomial has a certain root or factor

Word Problems

• Involve sharing, grouping, or measuring
• Examples:
  • Sharing: equal distribution among groups
  • Grouping: placing items into groups of a certain size
  • Measuring: filling containers of a certain capacity
• To solve, identify the total amount, number of groups, and number of items per group

Divisibility Rules

• Used to determine if a number is divisible by another without performing the actual division
• Common rules:
  • Divisible by 2: even last digit (0, 2, 4, 6, or 8)
  • Divisible by 3: sum of digits is divisible by 3
  • Divisible by 4: last two digits form a number divisible by 4
  • Divisible by 5: last digit is 0 or 5
  • Divisible by 6: divisible by both 2 and 3
  • Divisible by 9: sum of digits is divisible by 9
  • Divisible by 10: last digit is 0
• Allow quick determination of divisibility

Long Division

• A step-by-step procedure to divide a large number (dividend) by a smaller number (divisor)
• Dividend and divisor are written, and the first digit of the dividend is divided by the divisor
• The result is written below, and the divisor is multiplied by the result and subtracted from the dividend
• The next digit of the dividend is brought down, and steps 2-3 are repeated
• The process continues until the dividend is reduced to zero

Word Problems

• Division word problems involve sharing, grouping, or measuring quantities
• Examples include sharing cookies among friends, grouping books into sets, and measuring sugar for a recipe
• These problems require dividing a quantity into equal parts or groups

Division With Decimals

• Division with decimals follows the same procedure as whole number division
• The only difference is the presence of decimal points, which must be aligned
• Examples include dividing 4.5 by 1.5 and 12.75 by 3.25

Real-world Applications

• Measuring ingredients for cooking or baking
• Sharing resources or materials among a group
• Calculating rates or ratios, such as miles per hour or cost per unit
• Solving problems involving area or volume

Remainders

• The remainder is the amount left over after dividing one number by another
• Examples include 17 ÷ 5 with a remainder of 2 and 23 ÷ 4 with a remainder of 3
• Remainders are useful in real-world applications, such as finding leftover items or calculating material needed