Mathematics
Grade 9
15 min
Division with decimal quotients
Division with decimal quotients
What you'll learn
- Identify which congruence theorem (SSS, SAS, ASA, AAS) can be used to prove the congruence of two triangles, given a diagram or written description showing the side lengths and angle measures, with 80% accuracy.
- Apply the SSS, SAS, ASA, and AAS congruence theorems to solve for unknown side lengths or angle measures in geometric diagrams involving congruent triangles, achieving a minimum score of 70% on a problem set.
- Explain, in a written paragraph, the logical reasoning behind why the SSS, SAS, ASA, and AAS theorems are valid methods for proving triangle congruence, using precise mathematical vocabulary and demonstrating comprehension of the underlying principles, as assessed by a rubric.
- Construct a two-column proof to demonstrate the congruence of two triangles using SSS, SAS, ASA, or AAS, given a diagram and sufficient information, with all steps logically justified and accurately cited.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Perform long division with whole numbers to produce a decimal quotient.
Set up a polynomial long division problem correctly, including placeholders for missing terms.
Apply the division algorithm (Divide, Multiply, Subtract, Bring Down) to divide a polynomial by a binomial.
Identify the polynomial quotient and the remainder from the division process.
Express the result of polynomial division as a mixed rational expression in the form Q(x) + R(x)/D(x).
Substitute a given value for the variable to evaluate the rational expression and find a final decimal quotient.
Ever wonder how a GPS calculates the exact time to a destination, even with fractional minutes? It involves precise division! 🗺️
This tutorial connects the long division you already know with pol...
2
Key Concepts & Vocabulary
TermDefinitionExample
Rational ExpressionA fraction where both the numerator (top) and the denominator (bottom) are polynomials. The denominator cannot be zero.(x^2 + 2x + 1) / (x - 3)
DividendThe polynomial that is being divided.In (x^2 + 5x + 6) ÷ (x + 2), the dividend is x^2 + 5x + 6.
DivisorThe polynomial that you are dividing by.In (x^2 + 5x + 6) ÷ (x + 2), the divisor is x + 2.
QuotientThe main result of the division, before considering the remainder.In 25 ÷ 4, the quotient is 6.
RemainderThe value or polynomial 'left over' after the division is complete.In 25 ÷ 4, the remainder is 1.
Decimal QuotientThe complete numerical result of a division, including the fractional part expressed as a decimal.The decimal quotient of 25 ÷ 4 is 6.25.
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Core Formulas
The Division Algorithm
P(x) = D(x) \cdot Q(x) + R(x)
This states that any polynomial Dividend, P(x), can be rewritten as its Divisor, D(x), times its Quotient, Q(x), plus its Remainder, R(x). This is the foundation of the long division process.
Rational Expression Form
\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}
This rule shows how to write a rational expression using the results of polynomial long division. To find a final decimal value, you evaluate this expression for a given value of x.
Long Division Cycle
1. Divide -> 2. Multiply -> 3. Subtract -> 4. Bring Down
This is the repeating four-step process used in both numerical and polynomial long division to find the quotient and remainder.
4 more steps in this tutorial
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Challenging
The area of a rectangular field is given by the polynomial A(x) = 2x³ + 7x² + 2x - 3. The length is given by L(x) = 2x - 1. Find the width, W(x), and then calculate the numerical value of the width when x = 1.5.
A.9.75
B.10.5
C.11.25
D.12.0
Challenging
For what value of 'k' will the polynomial (x³ - 4x² + kx - 10) have a remainder of 4 when divided by (x - 2)?
A.5
B.7
C.9
D.11
Challenging
Let P(x) = 4x³ - 8x² + 5 and D(x) = 2x² - 3. Find the expression P(x)/D(x) in the form Q(x) + R(x)/D(x). Then, calculate the decimal quotient for x = -1.5, rounded to two decimal places.
A.-17.67
B.-10.33
C.-7.00
D.3.67
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Frequently asked questions
What grade level is "Division with decimal quotients"?
Division with decimal quotients is a Grade 9 Mathematics lesson on ExcelOS.
What will I learn in Division with decimal quotients?
You'll be able to: Identify which congruence theorem (SSS, SAS, ASA, AAS) can be used to prove the congruence of two triangles, given a diagram or written description showing the side lengths and angle measures, with 80% accuracy; Apply the SSS….
Is "Division with decimal quotients" free to practice?
Yes. You can read the tutorial preview for free, and signing up for a free ExcelOS account unlocks the full tutorial and all practice questions with instant feedback.
How many practice questions are included with Division with decimal quotients?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.