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Polynomials and FOIL
Polynomial questions look like alphabet soup, but they test four moves: combine what matches, multiply every term, FOIL two brackets, and run it all backwards to factor. Each move has one trap.
Multiplying is finding an area
| × | x | + 6 |
|---|---|---|
| x | x² | 6x |
| + 4 | 4x | 24 |
A rectangle that is x + 4 tall and x + 6 wide splits into four pieces. Add the pieces: x² + 6x + 4x + 24 = x² + 10x + 24. Every way of multiplying polynomials, FOIL included, is just a way of not missing a piece.
Like terms only
You can add 4x and 6x (both are x) to get 10x. You cannot add x² and 10x: they are different pieces, like square tiles and long tiles. They stay separate.
How to solve them
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1
Add and subtract only like terms
Like terms have the same letter with the same exponent. Line them up and add their numbers. When you subtract a bracket, the minus sign changes the sign of every term inside it.
Next step: DistributeAdding(3x² + 5x − 2) + (x² − 2x + 7)
4x² + 3x + 5
Subtracting(5x + 4) − (2x − 3)
5x + 4 − 2x + 3 = 3x + 7
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2
The distributive property: multiply every term
A number outside a bracket multiplies each term inside, including the sign. A negative number outside flips every sign.
Next step: FOIL- −3(2x − 5)
- −3 × 2x − 3 × (−5)
- −6x + 15
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3
FOIL: First, Outer, Inner, Last
To multiply two brackets of two terms each, make the four products in the same order every time, then combine the two middle terms.
Firstx · xx²Outerx · 66xInner4 · x4xLast4 · 624Next step: Factor out- (x + 4)(x + 6)
- x² + 6x + 4x + 24
- x² + 10x + 24
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4
Factoring out the greatest common factor
Find the biggest number and the highest power of x that go into every term, and write it in front of a bracket. Check by distributing it back.
Next step: Factor x² + bx + c- 12x² + 18x
- 6x · 2x + 6x · 3
- 6x(2x + 3)
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5
Factor x² + bx + c: two numbers that multiply to c and add to b
FOIL backwards. Look for two numbers whose product is the last number and whose sum is the middle number.
Next step: Special productsExampleFactor x² + 9x + 20
- Pairs that multiply to 20: 1 and 20, 2 and 10, 4 and 5.
- Only 4 and 5 add up to 9.
- x² + 9x + 20 = (x + 4)(x + 5). Check: x² + 5x + 4x + 20.
Answer (x + 4)(x + 5)
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6
Two special products worth memorizing
A sum times a difference loses its middle terms. A bracket squared keeps them: it is not just the two squares.
Difference of squares(x + 7)(x − 7)
x² − 49 (the middle terms cancel)
A square(x + 3)²
x² + 6x + 9, not x² + 9
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Squaring term by term
(x + 5)²:
Wrongx² + 25
Rightx² + 10x + 25
The minus sign on the first term only
(6x + 2) − (x + 4):
Wrong5x + 6
Right5x − 2
Adding unlike terms
3x + 4x²:
Wrong7x³
Rightstays 4x² + 3x
Forgetting the middle terms
(x + 2)(x + 8):
Wrongx² + 16
Rightx² + 10x + 16
5 questions at test level
Pick an answer, then read why. No timer, no account.
Simplify: (4x² + 3x − 5) + (2x² − 7x + 1)
Show the answer
B. Add like terms: 4x² + 2x² = 6x², 3x − 7x = −4x, and −5 + 1 = −4. 6x² + 4x − 4 loses the sign of −7x. The sum is 6x² − 4x − 4.
Multiply: (x + 3)(x − 8)
Show the answer
A. FOIL: x · x = x², x · (−8) = −8x, 3 · x = 3x, 3 · (−8) = −24. The middle terms give −8x + 3x = −5x. x² − 24 forgets the middle terms. The product is x² − 5x − 24.
Factor completely: 15x³ − 10x²
Show the answer
B. The greatest common factor is 5 (of 15 and 10) times x² (the lower power): 15x³ − 10x² = 5x²(3x − 2). The other choices are correct but not complete: something common is left inside. The answer is 5x²(3x − 2).
Factor: x² − 2x − 15
Show the answer
C. Two numbers that multiply to −15 and add to −2: −5 and 3. So x² − 2x − 15 = (x − 5)(x + 3). Check: x² + 3x − 5x − 15. (x + 5)(x − 3) gives +2x. The answer is (x − 5)(x + 3).
If x = 3, what is the value of (x + 4)(x − 1)?
Show the answer
D. Put 3 in each bracket: (3 + 4)(3 − 1) = 7 × 2 = 14. There is no need to multiply out first. The value is 14.
10 flashcards
Tap a card to flip it. Keep the ones you miss until they stick.
All 10 cards known.
You finished polynomials and foil.
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