- Terms are separated by addition or subtraction at the outermost level. - Keep the subtraction sign with the term that follows it. - A coefficient is the numerical factor multiplying a variable expression.

a) The three terms are \(7x^2\), \(-3x\), and \(11\). b) The coefficient of \(x^2\) is \(7\), the coefficient of \(x\) is \(-3\), and the constant term is \(11\).

a) \(7x^2, -3x, 11\) b) Coefficient of \(x^2\): \(7\); coefficient of \(x\): \(-3\); constant: \(11\).

- First look at the expression from the outside rather than expanding it. - Factors are quantities joined by multiplication at the outermost level. - Parentheses can make a multi-part expression act as one quantity.

a) The two factors are \(6\) and \((x-4)^2\). b) The grouped expression \(x-4\) is treated as one quantity and squared.

a) \(6\) and \((x-4)^2\) b) \(x-4\)

- Ignore the operations inside the power temporarily and look at the whole expression from the outside. - A term includes the sign immediately attached to it.

1. The whole expression has the form \(9-[2(y+3)^4]\), so subtraction is the operation combining the two outermost terms. 2. The two terms are \(9\) and \(-2(y+3)^4\).

a) Subtraction b) \(9\) and \(-2(y+3)^4\)

- Find the part that contains \(t\). - Include the minus sign when naming the constant term.

1. The term with the variable is \(7t\), whose coefficient is \(7\). 2. The term without the variable is \(-9\).

Variable term: \(7t\); coefficient: \(7\); constant term: \(-9\).

- Read adjacent letters as multiplication. - A constant term would stand alone in a sum or difference.

1. The coefficient is \(5\), and the variable factors are \(a\) and \(b\). 2. There is no separate constant term added or subtracted.

Coefficient: \(5\); variable factors: \(a\) and \(b\); no separate constant term.

- Separate the expression at addition and subtraction signs. - Try substituting \(x=2\) to see why the two variable terms play different roles.

1. The terms are \(4x^2\), \(3x\), and \(-6\), so there are three. 2. The coefficient of the \(x\) term is \(3\).

Three terms; coefficient of \(x\) is \(3\).

- Which operation links the \(3\) to the rest of the expression? - If you temporarily replaced the parentheses and everything inside them with one letter, what would the expression look like?

1. The expression is the product of \(3\) and the sum \(2+x\). 2. Parentheses group that sum so it can be treated as one factor.

The factors are \(3\) and \(2+x\). Parentheses keep the sum together.

- Find the expression directly after the subtraction sign. - Check where its parentheses end.

1. The parentheses group \(a+b\) as one quantity. 2. The whole sum \(a+b\), not just \(a\), is subtracted from \(12\).

The whole quantity \(a+b\) is subtracted; no, it is not just \(a\).

- What part of the bill remains if no games are played? - Compare the total for one game with the total for two games. What changed?

1. The term \(2g\) gives the total cost of the games, separate from the lane fee. 2. Each game costs \(\$2\), since the charge is \(2\) times the number of games.

\(2g\) is the charge for the games, without the lane fee. Each game costs \(\$2\).

- Match each variable with the item it counts. - A coefficient tells how much one unit contributes to the total. - A constant does not change when either variable changes.

a) The term \(8p\) is the total cost, in dollars, of the \(p\) posters. b) The coefficient \(3\) is the cost in dollars of one button pack. c) The constant \(25\) is the one-time setup charge in dollars.

a) The poster cost. b) \(\$3\) per button pack. c) The \(\$25\) one-time setup charge.

- Identify operations at the outermost level before looking inside parentheses. - A term can itself be a product of several factors. - Parentheses can signal that an entire subexpression is being used as one object.

a) The two outermost terms are \(4(2x-5)^2\) and \(9\). b) Two factors of the first term are \(4\) and \((2x-5)^2\). c) The quantity \(2x-5\) is formed first and then the entire quantity is squared. It can therefore be treated as one input to the squaring operation.

a) \(4(2x-5)^2\) and \(9\) b) \(4\) and \((2x-5)^2\) c) \(2x-5\) is one grouped subexpression that is squared as a whole.

- Start at the lowest operation in the tree and follow the outputs upward. - Use parentheses when the output of one operation becomes a single input to another. - For the final terms, look only at the outermost addition or subtraction.

a) The tree first forms \(2x\), then adds \(3\), multiplies that result by \(4\), and finally subtracts \(7\). The expression is \(4(2x+3)-7\). b) The subexpression completed before multiplying by \(4\) is \(2x+3\). c) The two outermost terms are \(4(2x+3)\) and \(-7\).

a) \(4(2x+3)-7\) b) \(2x+3\) c) \(4(2x+3)\) and \(-7\)

- Separate the expression into the product of its two main factors. - Ask what multiplying an amount by \(1.03\) does to that amount. - Treat the whole power \((1.03)^t\) as one factor rather than interpreting its pieces separately first.

a) The factor \(500\) is the initial investment amount, in dollars. b) The number \(1.03\) is the yearly growth factor: after one year, the amount is multiplied by \(1.03\), corresponding to a \(3\%\) increase. c) The factor \((1.03)^t\) is the accumulated growth factor after \(t\) years. It tells how many times the initial amount is scaled after \(t\) yearly growth steps.

a) The initial \(\$500\) investment. b) A yearly growth factor of \(1.03\), corresponding to \(3\%\) growth per year. c) The total growth factor after \(t\) years.

- Build the innermost quantity before applying the square. - Keep the phrase “the square of the quantity” grouped with parentheses. - The negative constant forms its own outer term.

The quantity \(x-4\) is squared first, giving \((x-4)^2\). Multiplying by \(3\) gives the first term \(3(x-4)^2\). Adding the constant term \(-7\) gives \(3(x-4)^2-7\). Its two outermost terms are \(3(x-4)^2\) and \(-7\).

\(3(x-4)^2-7\); its terms are \(3(x-4)^2\) and \(-7\).

- Identify which part of the expression does not depend on \(m\). - Think about what subtracting \(3\) from the trip-meter reading counts. - Treat \(m-3\) as one quantity before interpreting the multiplication by \(2.50\).

1. The constant \(4\) is a fixed \(\$4\) charge that does not depend on miles beyond \(3\). 2. The quantity \(m-3\) counts miles beyond the first \(3\) miles. 3. The term \(2.50(m-3)\) is the charge, in dollars, for those additional miles at \(\$2.50\) per mile.

a) The fixed \(\$4\) charge. b) Miles beyond the first \(3\) miles. c) The mileage charge for those additional miles at \(\$2.50\) per mile.

- Follow each branch from its leaves to the top. - The top operation tells how the two completed branches are related. - A subexpression can contain terms and still act as one factor in a larger expression.

a) The left branch forms \(a+b\), and the right branch forms \(a-b\). The top operation multiplies these results, so the expression is \((a+b)(a-b)\). b) The two factors are \(a+b\) and \(a-b\). c) Each sum or difference is completed on its own branch before the multiplication occurs. Therefore, each entire binomial serves as one input, or one factor, in the final product.

a) \((a+b)(a-b)\) b) \(a+b\) and \(a-b\) c) Each binomial is formed first and then used as one input to the final multiplication.

- What operation connects the two side lengths of a rectangle to its area? - Use the factored form as written rather than expanding it. - What sign must a physical length have?

1. The area of a rectangle is the product of its side lengths. Because the area is written as \((x+4)(x-2)\), the two factors \(x+4\) and \(x-2\) can represent the two side lengths. 2. A side length must be positive. The condition \(x>2\) guarantees \(x-2>0\), and it also gives \(x+4>0\). 3. The whole product represents the total area of the rectangular garden, measured in square feet.

a) \(x+4\) feet and \(x-2\) feet b) It guarantees both expressions are positive, so they can represent lengths. c) The total area of the garden, in square feet

- Decide whether the fee applies to each ticket or to the order. - Say in words what multiplying the entire parentheses by \(4\) would mean.

1. Four tickets cost \(4t\) dollars. 2. Add the service fee once: \(4t+5\). 3. The other expression charges the fee four times, once per ticket.

\(4t+5\). The expression \(4(t+5)\) charges four service fees instead of one.

- Start with the deepest pair of grouping symbols. - Work outward to see the last operation.

1. The innermost subexpression is \(n-1\). 2. The whole quantity multiplied by \(2\) is \(3+(n-1)\). 3. After the additions and subtraction inside the grouping, the final operation is multiplication by \(2\).

Innermost: \(n-1\); whole group: \(3+(n-1)\); last operation: multiplication by \(2\).

- Decide which quantity counts tables that can actually be used. - Check what each proposed expression subtracts from the full-table total.

1. There are \(s-2\) usable tables, each needing \(5\) brushes. 2. The correct expression is \(5(s-2)\). 3. The expression \(5s-2\) subtracts only two brushes rather than accounting for two unusable tables.

\(5(s-2)\). The expression \(5s-2\) removes two brushes instead of the brushes for two tables.

- How do the two types of trip contribute to the total? - Compare the number of trips of each type before and after the change.

1. Canoe trips carry \(3r\) customers and kayak trips carry \(2(r+1)\), giving \(3r+2(r+1)\). 2. Raising \(r\) by \(1\) adds one canoe trip, or \(3\) more customers. 3. It also adds one kayak trip, or \(2\) more customers. The total increases by \(3+2=5\).

Total: \(3r+2(r+1)\) customers. Increasing \(r\) by \(1\) adds \(3\) canoe and \(2\) kayak customers, or \(5\) in all.

- Evaluate the parentheses before multiplying in the first expression. - Do the operations in the second expression in their written order.

1. Evaluate the original: \(3(2+4)=18\). 2. Evaluate Jamie's expression: \(3\cdot2+4=10\). 3. The values differ because \(3\) multiplies the whole sum \(x+4\).

At \(x=2\), the original is \(18\) but \(3x+4=10\). The \(3\) multiplies both parts of the sum.

- Which part of the rule tells you the amount at the start? - When the drain rate changes, which part of the expression changes? - Check that the new formula predicts less water at the same positive time.

1. At \(t=0\), \(W=40\) gallons. 2. At \(t=4\), \(W=40-3\cdot4=28\) gallons. 3. Doubling the loss per minute changes the coefficient from \(-3\) to \(-6\), but keeps the starting amount \(40\). 4. The new rule is \(W=40-6t\); after \(4\) minutes it gives \(W=16\) gallons.

a) \(40\) gallons; \(28\) gallons. b) \(W=40-6t\); \(16\) gallons after \(4\) minutes.

- Calculate what the money left would be if the game card costs \(\$5\). - Does the number \(22\) depend on the cost of the game card?

1. At \(m=5\), both forms give \(30-(8+5)=22-5=17\) dollars, not \(22\). 2. The \(22\) is the balance after paying \(\$8\) for the snack, before buying the game card.

At \(m=5\), \(30-(8+5)=22-5=17\), so the friend's claim is false. \(22\) is the balance after buying only the snack.

- Try \(x=2\) in each set of instructions. Must the two results agree? - Check that your symbols follow the order of the spoken steps.

1. The first phrase means \(3(10-x)\), because all of \(10-x\) is multiplied by \(3\). 2. The second means \(10-3x\), because only \(x\) is multiplied by \(3\).

First: \(3(10-x)\). Second: \(10-3x\). The parentheses make the entire difference one factor.

- What would multiplying the \(5\) by \(p\) mean in the garden? - Try a small number of rows to test whether the expression counts the tomatoes once or more than once.

1. There are \(pr\) corn plants. The \(5\) tomato plants are added once, giving \(pr+5\). 2. The incorrect expression \(p(r+5)=pr+5p\) treats the tomatoes as if each row had five more plants. 3. For \(p=4\) and \(r=6\), the incorrect count is \(4(6+5)=44\), while the correct count is \(4\cdot6+5=29\). It overcounts by \(15\).

The correct total is \(pr+5\), not \(p(r+5)\), because the five tomatoes are not in every corn row. At \(p=4\), \(r=6\), the wrong expression gives \(44\) instead of \(29\): \(15\) too many.

- Start with the operation at the top of the tree and follow its two branches. - Where would parentheses be needed if you wrote the calculation on one line?

1. The denominator branch adds \(a\) and \(3\), so the entire denominator is \(a+3\). 2. The complete expression is \(\frac{20}{a+3}\). 3. Dividing by just \(a\) would ignore the additional \(3\) in the denominator.

\(\frac{20}{a+3}\); the whole denominator is \(a+3\), not just \(a\).

- If \(x\) changes, which parts of the drawing would change length? - Think about how to get an area from two perpendicular lengths. - Does the expression in c) describe the entire drawing or only part of it?

1. The whole width is \(x+2\) because it joins pieces of widths \(x\) and \(2\). 2. The total area is height times whole width, or \(h(x+2)\). 3. The area of the narrower right-hand piece is \(2h\).

a) The whole rectangle's width. b) The whole rectangle's area. c) The area of the narrower right-hand piece.

- Look first at the operations outside all parentheses. - A factor can contain addition without becoming several outer terms. - Distinguish the structure of the whole expression from the structure inside one factor.

1. Salim split the expression inside the parentheses even though \(x+2\) is grouped and multiplied by \(6\). Terms of the entire expression are determined by addition or subtraction at the outermost level. 2. The two terms are \(6(x+2)\) and \(-5\). 3. The visible factors of the first term are \(6\) and \(x+2\).

a) Salim treated parts inside a grouped factor as outer terms. b) \(6(x+2)\) and \(-5\) c) \(6\) and \(x+2\)

- Read each form according to its outermost operation. - In the factored form, multiplication is the outermost operation. - Ask how multiplying \(x+3\) by \(6\) affects each part inside the parentheses.

a) The terms are \(6x\) and \(18\). The coefficient of \(x\) is \(6\), and the constant term is \(18\). b) The visible factors are \(6\) and \(x+3\). c) The factor \(6\) shows that both terms in the expanded form are multiples of \(6\): \(6x\) is \(6\) times \(x\), and \(18\) is \(6\) times \(3\).

a) Terms: \(6x\) and \(18\); coefficient \(6\); constant \(18\). b) Factors: \(6\) and \(x+3\). c) The factor \(6\) is common to both expanded terms.

- Temporarily replace every occurrence of the repeated parenthetical expression by one symbol. - Identify the outermost addition before analyzing each term internally. - Compare how the same grouped quantity is used in the two terms without expanding it.

1. With \(u=x+5\), the expression has the structure \(2u+3u^2\): one term is twice \(u\), and the other is three times the square of \(u\). 2. The two outermost terms are \(2(x+5)\) and \(3(x+5)^2\). 3. Both terms are built from the same quantity \(x+5\). Treating that quantity as one repeated entity makes the relationship between the terms visible: one uses the first power of the entity and the other uses its square.

a) \(2u+3u^2\), where \(u=x+5\) b) \(2(x+5)\) and \(3(x+5)^2\) c) The same subexpression \(x+5\) is the common building block of both terms, appearing once to the first power and once squared.

- In part a), can the same total be counted in different orders? - Does joining late change the earlier delivery? - Check the two proposed totals using \(x=1\) and \(y=1\).

1. The original business count is \(x+y\); counting by delivery gives \(2(x+y)+3(x+y)\). 2. Counting all menus per business gives \(5(x+y)\). Both totals are equal. 3. Only the afternoon delivery includes the added food truck, so the updated total is \(2(x+y)+3(x+y+1)=5(x+y)+3\). 4. The expression \(5(x+y+1)\) incorrectly gives the new food truck the \(2\) morning menus as well.

a) \(2(x+y)+3(x+y)\) and \(5(x+y)\). b) \(2(x+y)+3(x+y+1)=5(x+y)+3\). The expression \(5(x+y+1)\) counts \(2\) morning menus for the new food truck, although it joined only later.