Restoration Series · Quiz 39 · Out and Back

The Guide

Before you begin

A rectangle cut into four: the picture of two binomials multiplied together, and the one this quiz keeps returning to.

This is the first quiz on this site for family 11, and it picks up where the app Polynomials: Like With Like leaves off. That room taught you to read a polynomial — to cut it into terms, name the parts of a term, find the degree, and put like with like. If you have not done it, do it first; everything here stands on it.

What this quiz adds is three things, and they are really one thing seen from two directions.

You will add and subtract whole polynomials, which is gathering like terms with parentheses in the way. You will multiply two binomials together (a binomial is a polynomial with exactly two terms, such as x + 3), which is the “out” of the title. And you will factor, which is the “back” — taking a polynomial and finding the binomials it came from.

Factoring is the one to spend your time on. It is the hardest thing here, it is the thing most people skip, and it is the tool that family 12's quadratic equations are built out of. Doing it well now saves a great deal of trouble later.

A short recap, so we agree on words

A term is a number on its own, a variable on its own, or a number multiplied by one or more variables: 7, x, 5x, −3x². The plus or minus sign in front of a term belongs to it and travels with it.

Like terms have the same variable raised to the same power. 3x and 7x are like terms. 3x and 3x² are not, however much they look alike, because a variable and its square are different things — a length and an area, if you want a picture.

And the rule that follows: you can only add or subtract like terms. Three apples plus seven apples is ten apples. Three apples plus three crates is not six of anything; it is three apples and three crates, and that is as far as it goes. 3x + 3x² stays 3x + 3x².

Adding polynomials

Adding two whole polynomials is gathering like terms, with one extra step at the start: the parentheses come off. Parentheses are grouping symbols: they mark the terms inside them as one group, to be treated together. When the group is added or subtracted as a whole, the grouping has to be undone before the like terms can be gathered.

(3x² + 5x − 2) + (x² − 8x + 7)

A plus sign in front of parentheses changes nothing inside them, so the parentheses simply go:

3x² + 5x − 2 + x² − 8x + 7

Now gather. The squares: 3x² + x² = 4x². The plain x terms: 5x − 8x = −3x. The numbers: −2 + 7 = 5.

= 4x² − 3x + 5

Write the answer in the usual order, highest power first. That is a convention rather than a rule, but the test writes its answers that way and matching them is easier than arguing.

Subtracting polynomials, and the sign that catches everyone

Subtraction is the same job with one difference, and the difference is where nearly every mistake in this family lives.

A minus sign in front of parentheses changes the sign of every term inside them. Every one, not just the first.

(3x² + 5x − 2) − (x² − 8x + 7)

Take the second set of parentheses off carefully, flipping all three signs as you go:

3x² + 5x − 2 − x² + 8x − 7

Now gather as before: 3x² − x² = 2x²; 5x + 8x = 13x; −2 − 7 = −9.

= 2x² + 13x − 9

Why every sign flips. The minus in front of the parentheses is a −1 multiplying the whole group inside them, and a multiplier reaches every term inside — that is what the earlier rooms called distributing. So −(x² − 8x + 7) is −1 × x², then −1 × (−8x), then −1 × 7, giving −x² + 8x − 7. The middle term came out positive because a negative times a negative is positive, which is family 2's rule doing its job here. If you flip only the first sign and leave the rest, you get 2x² − 3x + 5 — a wrong answer that looks entirely reasonable, which is why it is the trap.

A habit worth building: before you gather anything, rewrite the whole line with the parentheses gone and the signs fixed. Two steps on paper instead of one in your head, and the error disappears.

A number outside the parentheses

One recap from Algebra Practice 1, because the next section is built on it. A number outside the parentheses multiplies everything inside them:

3(x + 4) = 3x + 12 2(3x − 7) = 6x − 14 x(x + 5) = x² + 5x

That last one is worth a second look. Multiplying a variable by itself gives its square, so x × x = x². This is the rule from family 10 — when you multiply powers of the same variable you add the exponents — in its smallest case.

Multiplying two binomials

Now both sides have more than one term, and the rule is the plainest one in algebra: every term in the first binomial multiplies every term in the second.

Two terms times two terms means four multiplications. Count them as you go.

(x + 3)(x + 5)

Every term in the first binomial times every term in the second. Four multiplications, and the two middle ones are like terms.

That is x² + 5x + 3x + 15, and the two middle terms are like terms, so they gather: 5x + 3x = 8x.

(x + 3)(x + 5) = x² + 8x + 15

You may meet the name FOIL for this — First, Outer, Inner, Last, the order the four multiplications are done in. It is a fine memory aid and it has one flaw worth knowing about: it only describes two terms times two terms. Hand it a trinomial, a polynomial with three terms, and it has nothing to say. Everything times everything is the same rule without the limit, so that is the one to learn.

The same thing as a picture

There is a drawing underneath this, and it is worth seeing once because it makes the four pieces obvious and it is where the word square in “x squared” comes from.

A rectangle whose height is x + 3 and whose width is x + 5 has an area of (x + 3)(x + 5). Cut it along the lines that separate the x part from the number part on each side, and you get four smaller rectangles.

The four pieces are the four multiplications, and their areas add to the area of the whole.

The four pieces are the four multiplications, and their areas add to the area of the whole. This is exactly how al-Khwarizmi worked in Baghdad around the year 820, and exactly what Like With Like means by squares, things and coins: x² is a square, 8x is eight strips, 15 is fifteen unit tiles.

Two patterns worth noticing

Multiply out (x + 3)(x + 5) and you get x² + 8x + 15. Notice where 8 and 15 came from: 3 + 5 = 8, and 3 × 5 = 15.

That is not a coincidence, and it is not special to those numbers:

(x + a)(x + b) = x² + (a + b)x + ab

The middle number is the sum of the two. The last number is their product. Read that sentence backward and you have the whole of factoring, which is the next section.

The second pattern is what happens when the two numbers are the same size with opposite signs:

(x + 3)(x − 3) = x² − 3x + 3x − 9 = x² − 9

The middle terms cancel each other out and vanish. What is left is a square minus a square, and this shape — the difference of squares — turns up often enough that recognizing it saves real time.

The L-shaped piece, cut in two and laid out flat, is a rectangle (a + b) long and (a − b) tall.

Factoring: running the multiplication backward

Everything so far has started with terms grouped in parentheses and multiplied them out. Factoring goes the other way: you are handed x² + 8x + 15 and asked which two binomials it came from.

The pattern above tells you exactly what to look for. Two numbers that multiply to the last number and add to the middle number.

For x² + 8x + 15: two numbers that multiply to 15 and add to 8.

List the pairs that multiply to 15 — there are only two: 1 and 15, or 3 and 5. Check the sums: 1 + 15 = 16, no; 3 + 5 = 8, yes.

x² + 8x + 15 = (x + 3)(x + 5)

And then — every time, without exception — multiply your answer back out and check that you get what you started with. Factoring is the one piece of algebra where you can always check your own work completely, and it costs ten seconds.

The signs, which is where the thinking is

Finding the pair that multiplies right is arithmetic. Getting their signs right is the part that needs a moment's thought, and there are only three cases.

What the polynomial looks like	The two numbers	Example
Last number positive, middle positive	Both positive	x² + 7x + 12 = (x + 3)(x + 4)
Last number positive, middle negative	Both negative	x² − 7x + 12 = (x − 3)(x − 4)
Last number negative	One of each — and the larger one takes the sign of the middle term	x² + x − 12 = (x + 4)(x − 3) x² − x − 12 = (x − 4)(x + 3)

The reasoning behind the table is just the signs from family 2. If the two numbers multiply to something positive, they must have matched signs — both plus or both minus — and then the middle term tells you which. If they multiply to something negative, their signs must differ, and the one further from zero decides which way the middle term leans.

You do not have to memorize the table. You do have to check your signs by multiplying back out, and if you do that faithfully the table becomes something you notice rather than something you learned.

Look for a common factor first

Before anything else in a factoring problem, ask one question: is there something in every term?

3x² + 6x has a 3 in both terms and an x in both terms, so 3x comes out of each:

3x² + 6x = 3x(x + 2)

2x² + 10x + 12 has a 2 in every term. Take it out and what is left factors further:

2x² + 10x + 12 = 2(x² + 5x + 6) = 2(x + 2)(x + 3)

Two numbers multiplying to 6 and adding to 5: 2 and 3.

What “factor completely” means. A question that says factor completely is telling you to keep going until nothing can come out of anything. (2x + 4)(x + 3) does multiply back to 2x² + 10x + 12, so it is not wrong — but it is not finished, because 2x + 4 still has a 2 in both terms. Taking that 2 out gives 2(x + 2)(x + 3), and now nothing more can come out. Question 9 is about this.

Factoring a difference of squares

When there is no middle term at all and two square numbers are subtracted, the pattern from earlier runs backward:

x² − 9 = (x + 3)(x − 3) x² − 25 = (x + 5)(x − 5)

Take the square root of each part, and write the two binomials with a plus in one and a minus in the other. Recognizing this shape is worth a point on the test, because working it the long way — two numbers multiplying to −9 and adding to 0 — gets there too, but slowly.

One caution: this works for a difference of squares and not for a sum. x² + 9 does not factor at all with ordinary numbers, and a question that seems to ask you to factor it is asking something else.

The guard

Five things.

Subtracting a group in parentheses flips every sign inside it. Rewrite the line with the parentheses gone before you gather anything.

Everything times everything. Two terms by two terms is four multiplications; count them.

Common factor first, every time, before looking for a pair of numbers.

Multiply back out to check. This is not optional advice. It is the reason factoring questions are free points once you can do them.

Squaring a binomial is not squaring its parts. (x + 3)² means (x + 3)(x + 3), which is x² + 6x + 9 — not x² + 9. Question 8 is about this one, and it is probably the single most common mistake in all of algebra.

Worked Examples

Three to study before you start

Example 1 · Subtracting a whole polynomial

Simplify (6x² − 4x + 9) − (2x² + 3x − 5).

First rewrite it with the parentheses gone. The first group is untouched. The second is being subtracted, so all three of its signs flip: +2x² becomes −2x², +3x becomes −3x, and −5 becomes +5.

6x² − 4x + 9 − 2x² − 3x + 5

Now gather like terms. Squares: 6x² − 2x² = 4x². The x terms: −4x − 3x = −7x. Numbers: 9 + 5 = 14.

= 4x² − 7x + 14

Two of the three signs in that second group changed direction, and a student in a hurry changes one. The rewrite step is what protects you.

Example 2 · Multiplying two binomials

Multiply (x − 2)(x + 7).

Four multiplications, and the signs come along with the terms. The first binomial holds x and −2.

x × x = x². x × 7 = 7x. −2 × x = −2x. −2 × 7 = −14.

That gives x² + 7x − 2x − 14, and the two middle terms gather: 7x − 2x = 5x.

= x² + 5x − 14

Check it against the pattern: the two numbers are −2 and 7. Their sum is 5, which is the middle term. Their product is −14, which is the last term. Both match, so the answer is right — and you have just done, in reverse, the thinking that factoring needs.

Example 3 · Factoring, signs and all

Factor x² − 3x − 18.

First, is there a common factor? 1, 3 and 18 share nothing but 1, and there is no x in the last term, so no. Move on.

Now: two numbers that multiply to −18 and add to −3. The last number is negative, so the two numbers have opposite signs, and the middle term is negative, so the larger of the two carries the minus.

The pairs that multiply to 18 are 1 and 18, 2 and 9, 3 and 6. Their differences are 17, 7 and 3. We want 3, so the pair is 3 and 6, with the 6 taking the minus sign: −6 and +3.

Check: −6 × 3 = −18 and −6 + 3 = −3. Both right.

= (x − 6)(x + 3)

And multiply back out to be sure: x² + 3x − 6x − 18 = x² − 3x − 18. It matches. Notice that when the signs differ you are looking at the difference of the pair rather than the sum, which is often quicker to scan.

The Quiz · Ten Questions

Now you

Work on paper. No calculator is needed. Write every answer with the highest power first, and check every factoring answer by multiplying it back out. Questions 6 and 10 have two parts.

Simplify: (4x² + 3x − 5) + (2x² − 7x + 1)
Simplify: (5x² − 2x + 8) − (3x² + 6x − 4)
Multiply out: (x + 4)(x + 6)
Multiply out: (x + 5)(x − 2)
- A) x² − 3x − 10
- B) x² + 3x − 10
- C) x² + 3x + 10
- D) x² + 7x − 10
Factor: x² + 9x + 20
Factor each of these. (a) 3x² + 12x (b) x² − 16

Reading A reading rest stop, for the stubborn ones.

Ten minutes with the last section of this page — The Company, below the answer key — before you finish the quiz. It is about the man who invented the equals sign, wrote his mathematics books in English instead of Latin so that ordinary people could read them, and died in a debtors' prison for accusing an earl of theft. It will not help you with question 7. It is about why the symbols on this page look the way they do.

No photograph at this rest stop. The drawings in this quiz are all above.

Factor: x² − 5x − 14
Amara writes (x + 3)² = x² + 9. Is she right? If not, say what it really equals and explain in a sentence what went wrong.
Factor completely: 2x² + 10x + 12
- A) (2x + 4)(x + 3)
- B) 2(x + 2)(x + 3)
- C) (x + 2)(x + 6)
- D) 2(x + 1)(x + 6)
Two parts, and the second one uses the first. (a) Multiply out (x + 7)(x − 7). (b) Now factor x² − 100.

Answer Key

Check your work

1 6x² − 4x − 4

A plus in front of parentheses changes nothing, so the parentheses just come off: 4x² + 3x − 5 + 2x² − 7x + 1. Then gather. Squares: 4x² + 2x² = 6x². The x terms: 3x − 7x = −4x. Numbers: −5 + 1 = −4.

2 2x² − 8x + 12

All three signs in the second group flip: 5x² − 2x + 8 − 3x² − 6x + 4. Then gather: 5x² − 3x² = 2x²; −2x − 6x = −8x; 8 + 4 = 12. If you got 2x² + 4x + 4 you flipped the first sign and left the other two, which is the error this question is looking for. Notice in particular that −4 became +4: subtracting a negative adds.

3 x² + 10x + 24

Four multiplications: x×x = x², x×6 = 6x, 4×x = 4x, 4×6 = 24. Gather the middle: 6x + 4x = 10x. Or straight from the pattern: 4 + 6 = 10 for the middle, 4 × 6 = 24 for the last.

4 B) x² + 3x − 10

The two numbers are +5 and −2. Their sum is 3 and their product is −10. D) 7x adds 5 and 2 while ignoring the minus sign on the 2 — the signs travel with the numbers, always. A) has the middle sign backward, which is what you get by pairing −5 with +2 instead. C) gets the middle right but then makes the last term +10, forgetting that a positive times a negative is negative. Three wrong answers, three different sign slips, and all three are caught by the same habit: write the signs into the numbers before you start multiplying.

5 (x + 4)(x + 5)

Two numbers multiplying to 20 and adding to 9. The pairs for 20 are 1 and 20, 2 and 10, 4 and 5. Their sums are 21, 12 and 9. The last one is what we want. Both the last and the middle term are positive, so both numbers are positive. Check by multiplying back: x² + 5x + 4x + 20 = x² + 9x + 20.

6 (a) 3x(x + 4) (b) (x + 4)(x − 4)

(a) Both terms have a 3 and both have an x, so 3x comes out of each. 3x² ÷ 3x = x, and 12x ÷ 3x = 4, so what is left inside is x + 4. Check: 3x(x + 4) = 3x² + 12x. (b) A difference of squares: x² is x squared and 16 is 4 squared, and they are subtracted, so the binomials are (x + 4) and (x − 4). Check: x² − 4x + 4x − 16 = x² − 16, the middle terms cancelling as they always do in this pattern.

7 (x − 7)(x + 2)

Two numbers multiplying to −14 and adding to −5. The last number is negative, so the signs differ; the middle is negative, so the larger number takes the minus. Pairs for 14: 1 and 14, 2 and 7. Their differences are 13 and 5, and we want 5, so it is 7 and 2 with the 7 negative: −7 and +2. Check: −7 × 2 = −14 and −7 + 2 = −5. Multiplying back out gives x² + 2x − 7x − 14 = x² − 5x − 14.

8 No. (x + 3)² = x² + 6x + 9. She squared each part separately instead of multiplying the whole binomial by itself.

The small 2 applies to the whole binomial, everything inside the parentheses, so (x + 3)² means (x + 3)(x + 3). That is four multiplications: x², 3x, 3x and 9, which gather to x² + 6x + 9. The 6x is the part Amara's answer loses, and it is not a small part. Test it with numbers, which is the fastest way to settle any argument of this kind: let x = 2. Then (2 + 3)² = 5² = 25. Amara's version gives 4 + 9 = 13. The correct expansion gives 4 + 12 + 9 = 25. One number is enough to show which is right. The same mistake in its general form is thinking that squaring a sum is the sum of the squares, and it is worth being suspicious of that whenever a group in parentheses has a power on it.

9 B) 2(x + 2)(x + 3)

Take the common factor of 2 out first: 2(x² + 5x + 6). Then two numbers multiplying to 6 and adding to 5 are 2 and 3, giving 2(x + 2)(x + 3). Now the interesting part. A) also multiplies back to 2x² + 10x + 12 — check it and see — so A is not a wrong factorization. It is an unfinished one: 2x + 4 still has a 2 that can come out, and taking it out turns A into B. That is what the word completely is doing in the question. C) multiplies to x² + 8x + 12, which is not the same expression at all — the 2 has gone missing. D) multiplies to 2x² + 14x + 12, so its pair adds to 7 rather than 5. Two of these three wrong answers fail the check; the third fails only the word completely, and that is worth knowing about before you meet it on a test.

10 (a) x² − 49 (b) (x + 10)(x − 10)

(a) x² − 7x + 7x − 49, and the middle terms cancel, leaving x² − 49. (b) Part (a) showed you the shape. 100 is 10 squared, so x² − 100 is a difference of squares and factors as (x + 10)(x − 10). The two parts are the same fact traveling in opposite directions, which is what the title of this quiz is about: you multiply out to see what a thing becomes, and you factor to see what it was made of.

Reading your results

Eight to ten: you are ready for quadratic equations, which is family 12 and which runs entirely on the factoring you have just done. Five to seven: look at which kind you missed. Sign errors in 1, 2 or 4 mean the habit to build is rewriting the line with the parentheses removed before gathering anything. Trouble in 5, 7 or 9 means practicing the pair-finding on its own: take any two numbers, multiply and add them, and get used to going backward from the results. Under five: go back to Polynomials: Like With Like and then do questions 1, 2 and 3 again. Those three are the foundation, and there is no way through the rest of this family without them.

The Company · An Interlude

The man who made the equals sign

Every line of working on this page uses a symbol invented by one person on a particular afternoon, for a stated reason, and we know who and why.

In 1557 a Welsh physician named Robert Recorde published a book on algebra called The Whetstone of Witte. He was tired of writing out the words is equal to, over and over, in every line of every calculation. So he drew a pair of short parallel lines and said he would use them instead — a pair of what he called gemowe lines, from an old word for twins — and he gave his reason in the same sentence: “bicause noe .2. thynges, can be moare equalle.”

Nothing can be more alike than two lines of the same length lying side by side. That was the whole argument, and it was good enough. His lines were much longer than ours; over the next century printers shortened them, and they became the sign every schoolchild in the world now learns.

The rest of Recorde's life is the part worth the ten minutes.

He was born in Tenby, on the Welsh coast, around 1512. He studied at Oxford and then took a medical degree at Cambridge, and he practiced as a physician. Mathematics was the other half of his working life, and he did something with it that almost nobody was doing: he wrote in English.

In the 1540s and 1550s the learned books were in Latin, and Latin was the fence around the subject. If you had not been to a grammar school you could not get in. Recorde wrote The Grounde of Artes on arithmetic in 1543, The Pathway to Knowledg on geometry in 1551, The Castle of Knowledge on astronomy in 1556, and The Whetstone of Witte on algebra in 1557 — all of them in plain English, and all of them written as conversations between a master and a pupil, with the pupil interrupting to say he does not follow.

That form was a choice. A dialogue lets the learner be confused out loud and lets the confusion be answered, and Recorde used it for a reader he had clearly pictured: the merchant, the surveyor, the ship's officer, the apprentice. The Grounde of Artes was still being reprinted more than a hundred years after his death.

He also worked for the Crown. He was made comptroller of the mint at Bristol in 1549, and two years later was put in charge of the silver mines and the money in Ireland. In those posts he came up against William Herbert, the Earl of Pembroke, one of the most powerful men in England, and Recorde accused him of misconduct over the handling of the mines and the coinage.

Pembroke sued him for libel. Pembroke won. The damages were set at a thousand pounds, an impossible sum, and Recorde could not pay. He was committed to the King's Bench prison in Southwark, and he died there in 1558, having made his will in June of that year.

What happened between him and Pembroke is not fully knowable at this distance. What is on the record is the shape of it: a mathematician from Tenby accused an earl of theft from the public purse, the earl sued, and the mathematician died in prison.

Two things to keep. The first is that the symbols are not natural objects. Somebody made each of them, for a reason, usually to save work. The equals sign is two lines because two lines of the same length are the plainest picture of sameness anybody could think of — and knowing that is a small defence against the feeling that algebra is a code invented to keep you out. The second is that the man who wrote the first English algebra, so that people without Latin could learn it, spent his last months in a debtors' prison. Making knowledge available to everyone has never been a safe occupation, and it was not in 1557 either.

Sources: Recorde's The Whetstone of Witte (London, 1557) for the equals sign and his reason for it; the standard accounts of his life for the Bristol mint, the Irish mines, the suit brought by the Earl of Pembroke, and his death in the King's Bench prison in 1558.

Where this goes. This quiz has given family 11 its three missing pieces, and it has given family 12 the tool it runs on. Next in the series is quadratic equations — solving x² + 8x + 15 = 0 by factoring it into (x + 3)(x + 5) = 0 and reading the answers off — along with the quadratic formula for the ones that will not factor. The family pages for these are Family 11 and Family 12.