Multiplying polynomials · 마틴과 코딩하기

Expanding brackets and multiplying two pairs of brackets: each with each, skip nothing.

Each with each

You can add polynomials. Today you will multiply them. The basic move has a name: expanding: a number or a term in front of the brackets is multiplied with every term inside.

3 · (x + 4) = 3x + 12. The three visited both terms. It was not allowed to skip either.

Why does it work? 3 · (x + 4) means three packs, each with x + 4. Three packs = three x and three fours. Plain sense, no magic.

Today’s second step: two pairs of brackets. (x + 2)(x + 3) is multiplied in the style “each with each” — every term of the first pair with every term of the second. Four products appear, then they are collected.

Expanding is a craft. It does not need an idea, it needs order: arrows, mid-steps, a check. Anyone who draws arrows does not skip a term. Anyone who calculates from their head skips.

Sofie is calculating the area of a garden they want to extend. Now it measures x metres in width and x + 3 in length. After the extension the width will be x + 2. “The area will be (x + 2) · (x + 3),” she writes. Adam peeks: “And how much is that?” Sofie draws a rectangle and cuts it with lines into four fields: x · x, x · 3, 2 · x and 2 · 3. “Four pieces: x² + 3x + 2x + 6. I collect: x² + 5x + 6.” Adam looks at the picture: “So multiplying brackets is just a sliced rectangle?” Exactly.

💡

Draw arrows from term to term. With (x + 2)(x + 3) four arrows and four products must appear. Three products = an arrow is missing somewhere.

How to expand step by step

A term times brackets:

Draw arrows from the term in front of the brackets to every term inside.
Multiply along the arrows: 5 · (2x − 3) = 5 · 2x + 5 · (−3) = 10x − 15.
The signs travel with the terms: minus three times five is minus fifteen.

With a letter it runs the same: x · (x + 4) = x² + 4x. Remember: x · x = x².

Brackets times brackets — each with each:

(x + 2)(x + 3):

The first term of the first pair with both in the second: x · x = x², x · 3 = 3x.
The second term of the first pair with both in the second: 2 · x = 2x, 2 · 3 = 6.
Write all four: x² + 3x + 2x + 6.
Collect like terms: x² + 5x + 6.

A check question at the end: binomial times binomial = four products before collecting. Trinomial times binomial = six. Number of products = number of terms times number of terms.

Example 1: 4(3x − 5)

Arrows from the four to both terms:

4 · 3x = 12x and 4 · (−5) = −20.

Result: 12x − 20.

Check by substituting x = 2: the question 4 · (6 − 5) = 4 · 1 = 4. The result 24 − 20 = 4. It fits. This check takes ten seconds and catches almost every sign mistake.

Example 2: (x + 5)(x − 2)

Each with each, four products:

x · x = x², x · (−2) = −2x, 5 · x = 5x, 5 · (−2) = −10.

I write: x² − 2x + 5x − 10.

I collect the middle terms: −2x + 5x = 3x.

Result: x² + 3x − 10.

I carried the signs with the terms the whole time: minus two stayed minus two in both of its products.

Example 3: (2x + 3)(x + 4)

A coefficient on x changes nothing — still each with each:

2x · x = 2x², 2x · 4 = 8x, 3 · x = 3x, 3 · 4 = 12.

I write: 2x² + 8x + 3x + 12.

I collect: 8x + 3x = 11x.

Result: 2x² + 11x + 12.

Watch the first product: 2x · x = 2x², not 2x. Letters are multiplied too: x · x = x².

⚠️

The most common mistake: multiplying only the first term in the brackets. 3(x + 4) is not 3x + 4! The three must visit the four too: 3x + 12. Draw arrows until expanding sits in your hand.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 시작 with ten. When it goes well, add more.

Practise powers — when multiplying brackets you will meet x · x = x² in every question.

Practise the questions

On paper

Arrows, four products, collecting. Order beats speed.

Expand: 6(x + 2), 3(2a − 5), x(x + 7). Draw arrows for each one.
Multiply (x + 3)(x + 4). Write out all four products, then collect.
Multiply (x − 6)(x + 2) and check by substituting x = 1 into the question and the result.
A garden measures (x + 5) in length and (x + 1) in width. Write and expand the expression for its area.

Now you 💪

I multiply a term in front of the brackets with every term inside.
With two binomials I get four products.
I carry signs with the terms and check the result by substituting.

✅

Done when: You have expanded all the tasks with arrows and mid-steps, one check by substituting, and a run of five correct in practice.

What to take from this lesson

Expanding: the term in front of the brackets times every term inside.
Two pairs of brackets: each with each, then collect.
x · x = x² — letters are multiplied too.
Number of products = terms times terms. Fewer products = a mistake.
A check by substituting x = 1 or x = 2 catches most mistakes.