Polynomials: build, multiply, and check terms

Where we are on the map

Powered terms now combine into one expression.

Begin with a familiar image

If a rectangle’s side lengths are sums, its area is the sum of four smaller rectangle areas.

Precise meaning

A polynomial is a sum of terms with nonnegative integer powers. In multiplication, every term in one factor multiplies every term in the other.

The lesson’s support signal

meaning → model → calculation → check → explain in your own words

every term × every term → preserve signs → combine like terms → substitute to check

Worked example

(2x+3)(x−4)=2x²−8x+3x−12=2x²−5x−12.

Example with a fading prompt

Change the numbers in the example and repeat the solution without copying its steps. Estimate first, calculate exactly, and check against the original condition.

Recall without a prompt

1. Define the main idea in one sentence.
2. Rebuild the support signal from memory.
3. Solve the example with different numbers and explain why every step is valid.

Find and correct the mistake

Making the middle terms −8x−3x is wrong: 3×x is positive 3x, so they combine to −5x.

Transfer to a new setting

Apply the same relationship to something you can measure yourself. Name the quantities, units, and the boundary within which the model is valid. Check the answer by a second representation or an inverse operation.

Exercise

Required. Expand (3x−2)(x+5) and compare both forms at x=2.

With your own data. Create one everyday example with the same structure and show it in words, a formula, and a check.

Optional. Seven days later, change the numbers and repeat without the support map.

Open perspective

A quadratic function turns a degree-two polynomial into a parabola on a graph.

Course contents