Lesson Guide
Explanation
A binomial is an expression with two terms, like x plus three. To multiply two of them, draw a rectangle x plus three tall and x plus five wide, and split it into a two by two grid. The four boxes are x times x, which is x squared, x times five, which is five x, three times x, which is three x, and three times five, which is fifteen. Add the four pieces and collect the middle two: x squared plus eight x plus fifteen.
Your Turn
Expand (x + 2)(x + 7). Check: x2 + 9x + 14.
Watch Out
There are four pieces, not two. (x + 3)(x + 5) is not x2 + 15.
Explanation
Without a grid, multiply each term in the first bracket by each term in the second. Some people call the order first, outer, inner, last. For two x minus three, times x plus four: two x times x is two x squared, two x times four is eight x, negative three times x is negative three x, and negative three times four is negative twelve. Collect the middle terms: eight x minus three x is five x. The answer is two x squared plus five x minus twelve. Check it with x equals one: the brackets give negative one times five, which is negative five, and so does the answer.
Your Turn
Expand (3x + 1)(x − 2). Check: 3x2 − 5x − 2.
Watch Out
The sign stays with its term: −3 × 4 is −12, not +12.
Explanation
When both brackets have a minus sign, watch the last piece. For x minus six, times x minus two: x times x is x squared, x times negative two is negative two x, negative six times x is negative six x, and negative six times negative two is positive twelve. The middle terms are both negative, so they make negative eight x. The answer is x squared minus eight x plus twelve.
Your Turn
Expand (y − 4)(y − 5). Check: y2 − 9y + 20.
Watch Out
The constant is +12, not −12: negative × negative is positive.
Explanation
When the letters have numbers in front, the same four multiplications work. Three a minus two, times two a plus five. Three a times two a is six a squared. Three a times five is fifteen a. Negative two times two a is negative four a. Negative two times five is negative ten. Collect the middle: fifteen a minus four a is eleven a. The answer is six a squared plus eleven a minus ten.
Your Turn
Expand (2p + 3)(4p − 1). Check: 8p2 + 10p − 3.
Watch Out
Multiply the coefficients: 3a × 2a is 6a2, not 5a2 and not 6a.
Explanation
Sometimes a binomial product is only one part of the question. For x plus four, times x minus one, minus x times x plus two, expand the binomial product first: x squared plus three x minus four. Then expand x times x plus two and keep it in a bracket, because the minus sign in front belongs to all of it. Subtracting gives negative x squared and negative two x. Collect: the x squared terms cancel, and three x minus two x is x, so the answer is x minus four. Quiz and exercises now.
Your Turn
Expand and simplify (x + 1)(x + 3) − x(x + 4). Check: 3 — every x term cancels.
Watch Out
Subtract the whole product: −(x2 + 2x) is −x2 − 2x, not −x2 + 2x.
📌 Key Points — this lesson in a nutshell
- A binomial product (a + b)(c + d) has four pieces: ac + ad + bc + bd. Draw the 2 × 2 area grid if in doubt.
- Every term in the first bracket multiplies every term in the second. Carry each sign with its term.
- Collect the two middle pieces — they are usually like terms: (x + 3)(x + 5) = x2 + 8x + 15.
- Check by substituting a small number such as x = 1 into both the brackets and your answer.
Step 1: Introduction
Step 1 of 5
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