Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 9 chapters and video availability01 · Read and understand
What you will learn
- Use distribution to multiply two binomials and collect the result.
- Justify the method and check its domain, units, or logical conditions.
Before you start
Signed arithmetic, fraction and decimal operations, one-step equations, and introductory coordinate graphs.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Expand (x+3)(x+5).
Why this math matters
Use distribution to explain polynomial products instead of treating a mnemonic as a proof. The example connects a stated condition to a conclusion and then checks whether the result satisfies that condition.
Set up the model
A useful answer starts with clear assumptions:
- Use the real-number system and the restrictions or data model stated in the question unless another domain is specified.
- Treat provided measurements as exact classroom-model values unless an approximation or uncertainty is stated.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Multiply every pair of binomial terms
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Expand (x+3)(x+5).
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the conditions
x·x + x·5 + 3·x + 3·5
Each term of one factor multiplies each term of the other.
Develop the calculation
x² + 5x + 3x + 15
The four products are the four pieces of a positive-length area model when x>0.
Check and interpret
x²+8x+15
The algebraic identity holds for every real x even outside the area model's positive-length interpretation.
The result
x²+8x+15
The algebraic identity holds for every real x even outside the area model's positive-length interpretation.
Common mistakes to catch
- Do not omit the cross-products.
- A geometric model's domain can be narrower than the resulting algebraic identity.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Expand (y−2)(y+4).
Show a hint
Multiply all four term pairs.
Reveal answer and explanation
y²+2y−8
y²+4y−2y−8 combines to the result.
Practice 2
Is (x+3)² equal to x²+9?
Show a hint
The two cross-products must be included.
Reveal answer and explanation
No; it is x²+6x+9
Both x·3 and 3·x contribute to the middle term.
Take the idea with you
Use distribution to explain polynomial products instead of treating a mnemonic as a proof.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Reverse a product to solve a quadratic
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