\begin{equation*}
\blert{\text{Profit = Revenue - Cost}}
\end{equation*}
Section 7.5 Chapter 7 Summary and Review
Subsection Lesson 7.1 Polynomials
- A polynomial is a sum of terms, each of which is a power of a variable with a constant coefficient and a whole number exponent.
- The degree of a polynomial in one variable is the largest exponent that appears in any term.
- Like terms are any terms that are exactly alike in their variable factors. The exponents on the variable factors must also match.
- To add two polynomials, we need only remove parentheses and combine like terms.
- To subtract polynomials, we change the sign of each term within parentheses, remove the parentheses, and combine like terms.
Subsection Lesson 7.2 Products of Polynomials
First Law of Exponents.
To multiply two powers with the same base, we add the exponents and leave the base unchanged. In symbols,\begin{equation*} \blert{a^m \cdot a^n = a^{m+n}} \end{equation*}- We use the distributive law to compute products of two or more polynomials.
Subsection Lesson 7.3 More About Factoring
Second Law of Exponents.
To divide two powers with the same base, we subtract the smaller exponent from the larger one, and keep the same base.- If the larger exponent occurs in the numerator, put the power in the numerator.
- If the larger exponent occurs in the denominator, put the power in the denominator.
In symbols,- \(\displaystyle \blert{\dfrac{a^m}{a^n}= a^{m-n}~~~~~~\text{if}~~~~n \lt m}\)
- \(\displaystyle \blert{\dfrac{a^m}{a^n}= \dfrac{1}{a^{n-m}}~~~~~~\text{if}~~~~n \gt m}\)
- The greatest common factor (GCF) is the largest factor that divides evenly into each term of the polynomial: the largest numerical factor and the highest power of each variable.
- When we represent the product of two binomials by the area of a rectangle, the products of the entries on the two diagonals are equal.
To Factor \(~ax^2+bx+c~\) Using the Box Method.
- Write the quadratic term \(ax^2\) in the upper left sub-rectangle, and the constant term \(c\) in the lower right.
- Multiply these two terms to find the diagonal product, \(D\text{.}\)
- List all possible factors \(px\) and \(qx\) of \(D\text{,}\) and choose the pair whose sum is the linear term, \(bx\text{,}\) of the quadratic trinomial.
- Write the factors \(px\) and \(qx\) in the remaining sub-rectangles.
- Factor each row of the rectangle, writing the factors on the outside. These are the factors of the quadratic trinomial.
- We should always begin factoring by checking to see if there is a common factor that can be factored out.
Subsection Lesson 7.4 Special Products and Factors
Squares of Binomials.
- \(\displaystyle \blert{(a+b)^2=a^2+2ab+b^2}\)
- \(\displaystyle \blert{(a-b)^2=a^2-2ab+b^2}\)
Difference of Two Squares.
\begin{equation*} \blert{(a+b)(a-b)=a^2-b^2} \end{equation*}Special Factorizations.
- \(\displaystyle \blert{a^2+2ab+b^2=(a+b)^2}\)
- \(\displaystyle \blert{a^2-2ab+b^2=(a-b)^2}\)
- \(\displaystyle \blert{a^2-b^2=(a+b)(a-b)}\)
Sum of Two Squares.
The sum of two squares, \(~a^2+b^2~\text{,}\) cannot be factored.
Subsection Review Questions
Use complete sentences to answer the questions.
- Explain why \(\dfrac{x}{2}+3\) is a polynomial but \(\dfrac{2}{x}+3\) is not a polynomial.
- A classmate says that \(\sqrt{2}x^2+3\sqrt{2}x +1\) is a polynomial, but another classmate disagrees. Who is correct? Explain.
- If a polynomial is written in descending powers of the variable and the first term has degree 5, what is the degree of the polynomial? Give an example.
- If a polynomial of degree3 is added to a polynomial of degree 2, what is the degree of the sum? Give an example.
- A classmate tells you that, depending on the binomials used, the sum of two binomials can have one, two, three, or four terms. Give an example of each.
- If a polynomial of degree 3 is multiplied by a polynomial of degree 2, what is the degree of the product? Give an example.
- How many terms are there in the square of a monomial? Of a binomial? Give examples.
- Which of the following can be factored? Give examples.
- sum of two squares
- difference of two squares
- Explain the difference between the sum of two squares and the square of a binomial. Give examples.
- When you are trying to factor a polynomial, what is the best way to start?
Subsection Review Problems
Exercises Exercises
1.
Write the polynomial in descending powers of the variable and state the degree of the polynomial.
- \(\displaystyle 1-\dfrac{x^2}{2}+\dfrac{x^4}{24}-\dfrac{x^6}{720}\)
- \(\displaystyle 10n^4+10^6n+10^8n^2\)
Answer.
- \(\dfrac{-1}{720}x^6+\dfrac{1}{24}x^4-\dfrac{1}{2}x^2+1 \text{,}\) degree 6
- \(10n^4+10^8n^2+10^6n \text{,}\) degree 4
2.
Which of the following are polynomials?
- \(\displaystyle -25y^{10}-1\)
- \(\displaystyle \sqrt{3}x+2\)
- \(\displaystyle \dfrac{4}{x^3}-\dfrac{3}{x^2}+\dfrac{1}{x}\)
- \(\displaystyle t^4+5t^3-2\sqrt{t}+3\)
Exercise Group.
For Problems 3–8, evaluate the polynomial for the given values of the variables.
3.
\(-16t^2+50t+5~~~~\text{for}~~t=\dfrac{1}{2}\)Answer.
\(26\)
4.
\(\dfrac{1}{2}n^2+\dfrac{1}{2}n~~~~\text{for}~~n=100\)5.
\(\dfrac{1}{6}z^3+\dfrac{1}{2}z^2+z+1~~~~\text{for}~~z=-1\)Answer.
\(\dfrac{1}{3} \)
6.
\(p^4+4p^3+6p^2+4p+1~~~~\text{for}~~p=-2\)7.
\(4x^2-12xy+9y^2~~~~\text{for}~~x=-3,~y=-2\)Answer.
0
8.
\(2R^2S~~~~\text{for}~~R=150,~S=0.01\)9.
Write as a polynomial: \(~~~x(3x[2x(x-2)+1]-4)\)
Answer.
\(6x^4-12x^3+3x^2-4x\)
10.
Write as a polynomial:
\begin{equation*}
a^2(a-3)-2a(a^2+2a^3)+(a-2)(a-3)
\end{equation*}
11.
Suppose you want to choose four items from a list of possible items. The number of different ways you can make your choice is given by the polynomial
\begin{equation*}
\dfrac{1}{24}n^4-\dfrac{1}{4}n^3+\dfrac{11}{24}n^2-\dfrac{1}{4}n
\end{equation*}
- How many different sets of four compact disks can be chosen from a collection with 20 compact disks?
- Of course you cannot choose four different items from a list of only 3 possible items. What do you get when you evaluate the polynomial for \(~n=3\text{?}\) \(~n=2\text{?}\) \(~n=1\text{?}\)
- Evaluate the polynomial for \(~n=4~\text{.}\) Explain why your answer makes sense in term of what the polynomial represents.
Answer.
- 4845
- 0, 0, 0
- 1; There is exactly one way to choose 4 items from a list of 4—choose all 4 items.
12.
The sum of the cubes of the first counting numbers is given by
\begin{equation*}
\dfrac{n^4}{4}+\dfrac{n^3}{2}+\dfrac{n^2}{4}
\end{equation*}
Find the sum of the cubes of the first five counting numbers.
Exercise Group.
For Problems 13–14, simplify by combining like terms.
13.
\(8p^2q-3pq^2-2pq^2+p^2q\)Answer.
\(9p^2q -5pq^2\)
14.
\(1.7m^3+2.6-0.3m-1.4m^2-1.2m^3+4.5m^2+1.1\)Exercise Group.
For Problems 15–16,add or subtract the polynomials.
15.
\((4b^3+2b^2-3b+7)-(-2b+3-b^3-7b)\)Answer.
\(5b^3+2b^2+6b+4\)
16.
\((8w^6-5w^4-3w^2)+(2w^4-8w^2+4)\)Exercise Group.
For Problems 17–22, add or subtract the polynomials.
17.
- \(\displaystyle 3x^2+x^2\)
- \(\displaystyle 3x^2\cdot x^2\)
Answer.
- \(\displaystyle 4x^2\)
- \(\displaystyle 3x^4\)
18.
- \(\displaystyle 5b^3-b^3\)
- \(\displaystyle 5b^3(-b^3)\)
19.
- \(\displaystyle 7a^4+a^6\)
- \(\displaystyle 7a^4 \cdot a^6\)
Answer.
- cannot be simplified
- \(\displaystyle 7a^{10}\)
20.
- \(\displaystyle 24^4-r^3\)
- \(\displaystyle 2r^4(-4^3)\)
21.
- \(\displaystyle \dfrac{3b^9}{9b^3}\)
- \(\displaystyle \dfrac{9b^3}{3b^9}\)
Answer.
- \(\displaystyle \dfrac{b^6}{3}\)
- \(\displaystyle \dfrac{3}{b^6}\)
22.
- \(\displaystyle \dfrac{4m^8}{m^8}\)
- \(\displaystyle \dfrac{m^4}{8m^4}\)
Exercise Group.
For Problems 23–24, multiply.
23.
\((5m^2n)(-6m^3n^2)\)Answer.
\(-30m^5n^3\)
24.
\(-7qr^2(2q^4-1)\)Exercise Group.
For Problems 25–26, divide.
25.
\(\dfrac{-21x^4y^3}{3xy^3}\)Answer.
26.
\(\dfrac{3a^2b}{6a^4b^3}\)Exercise Group.
For Problems 27–32, multiply.
27.
\(-2y(y-4)(y+3)\)Answer.
\(-2y^3+2y^2+24y\)
28.
\(6p^4(2p+1)(p-4)\)29.
\((d-2)(d^2-4d+4)\)Answer.
\(d^3-6d^2+12d-8\)
30.
\((2k+1)(k-2)(k+3)\)31.
\(9x^3y(4x-3y)^2\)Answer.
\(144x^5y-216x^4y^2+81x^3y^3\)
32.
\((a-3)^3\)Exercise Group.
For Problems 33–44, add, subtract, multiply, or divide the polynomials as indicated.
33.
\((3a^2-4a-7)-(a^2-5a+2)\)Answer.
\(2a^2+a-9\)
34.
\((5ab^2+6a^2b)-(3ab^2+5ab)\)35.
\(\dfrac{13c^2d}{26c^3d}\)Answer.
\(\dfrac{1}{2c} \)
36.
\(\dfrac{12m^4+4m}{4m}\)37.
\(7q62(8-7q^2-q^4)\)Answer.
\(56q^2-49q^4-7q^6\)
38.
\(2k^2(-3km)(m^3k)\)39.
\((9v+5w)(9v-5w)\)Answer.
\(81v^2-25w^2\)
40.
\((x^2+1)^2\)41.
\(-3p^2(p+2)(p-5)\)Answer.
\(-3p^4+9p^3+30p^2\)
42.
\(12rs^2(3r-s)(r+4s)\)43.
\((2x-3)(4x^2+6x+9)\)Answer.
\(8x^3-27\)
44.
\((3x+2)(3x+2)(3x+2)\)Exercise Group.
For the rectangle in Problems 45–48, find
- its perimeter,
- its area.
45.
Answer.
46.
47.
Answer.
48.
49.
Evaluate the expression below for \(~~a=-6.3,~b=-4.8,~c=5.2\)
\begin{equation*}
(4a-3b-2c)-(a+6b-5c)+(3a+9b-2c)
\end{equation*}
Answer.
\(-32.6\)
50.
Write a polynomial for the volume of a box whose width is 10 centimeters less than its length, and 2 centimeters more than its height. Let \(w\) represent the width of the box.
51.
Nova cosmetics sells \(140-2p\) cans of styling mousse each month if they charge \(p\) dollars per can.
- Write a polynomial for the company’s monthly revenue from mousse.
- Find the revenue if each can costs $4.
Answer.
- \(\displaystyle 140p-2p^2\)
- $528
52.
Newsday magazine surveyed 400 people on the question “Do you think the government is spending too much on defense?” They reported the following results: Of the college-educated respondents, 72% answered yes, and 48% of those without a college education answered yes. Suppose you would like to know how many of the 400 people surveyed answered yes. You will need to know how many of the 400 have a college education. Let \(x\) represent this unknown value. Write and simplify expressions in terms of for each of the following.
- How many of the people surveyed do not have a college education?
- How many of the college-educated respondents answered yes?
- How many of those without a college education answered yes?
- How many people total answered yes?
Exercise Group.
For Problems 53–64, decide whether the expression is an equation or a polynomial. If it is an equation, solve it. If it is a polynomial, factor it.
53.
\(2x^2+x-3=0\)Answer.
Equation. \(x=\dfrac{-3}{2} \text{,}\) \(1\)
54.
\(a^2-9\)55.
\(2x^2+x-3\)Answer.
Polynomial. \((2x+3)(x-1)\)
56.
\(a^2=9\)57.
\(2x^2+x=0\)Answer.
Equation. \(x=\dfrac{-1}{2} \text{,}\) \(0\)
58.
\(a-9=0\)59.
\(2x+3=0\)Answer.
Equation. \(x=\dfrac{-3}{2} \)
60.
\(a^2=9a\)61.
\(2x^3-2x\)Answer.
Polynomial. \(2x(x+1)(x-1)\)
62.
\(a^4-16\)63.
\(p^3-p=0\)Answer.
Equation. \(p=-1\text{,}\) \(0\text{,}\) \(1\)
64.
\(n(n-3)(n+3)=0\)Exercise Group.
For Problems 65–68, factor out the greatest common factor.
65.
\(12x^5-8x^4+20x^3\)Answer.
\(4x^3(3x^2-2x+5)\)
66.
\(9a^4b^2+6a^3b^3-3a^2b^4\)67.
\(30w^9-42w^4+54w^8\)Answer.
\(6w^4(5w^5-7+9w^4)\)
68.
\(45x^2y^2+18x^2y^3-27x^3y^3\)Exercise Group.
For Problems 69–71, factor out a negative monomial.
69.
\(-10d^4+20d^3-5d^2\)Answer.
\(-5d^2(2d^2-4d+1)\)
70.
\(-6m^3n-18m^2n+6mn\)71.
\(-vw^5-vw^4+vw^2\)Answer.
\(-vw^2(w^3+w^2-1)\)
Exercise Group.
For Problems 72–74, factor out the common binomial factor.
72.
\(7q(q-3)-(q-3)\)73.
\(-r^2(3r+2)+4(3r+2)\)Answer.
\((3r+2)(-r^3+4)\)
74.
\(5(x-2)-x^2(x-2)\)Exercise Group.
For Problems 75–90, factor completely.
75.
\(3z^2-12z\)Answer.
\(3z(z+2)(z-2)\)
76.
\(-4x^3y+8x^2y-4xy\)77.
\(a^4+10a^2+25\)Answer.
\((a^2+5)^2\)
78.
\(4x^8-64\)79.
\(2a^2b^6+32a^6b^2\)Answer.
\(2a^2b^2(b^4+16a^4)\)
80.
\(4p^2q^4+32p^3q^3+64p^4q^2\)81.
\(-2a^4b-4a^3b^2+30ab^3\)Answer.
\(-2a^2b(a+5b)(a-3b)\)
82.
\(15r^3s+39r^2s^3-18rs^4\)83.
\(2q^4+6q^3-80q^2\)Answer.
\(2q^2(q+8)(q-5)\)
84.
\(32-b^4-14b^2\)85.
\(x^2-3xy+2y^2\)Answer.
\((x-2y)(x-y)\)
86.
\(y^2-3by-28b^2\)87.
\(3a^2b+12ab^2+9b^3\)Answer.
\(3b(a+3b)(a+b)\)
88.
\(80t^4-28t^3-24t^2\)89.
\(9x^2y^2+3xy-2\)Answer.
\((3xy+2)(3xy-1)\)
90.
\(4a^3x-2a^2x^2-12ax^3\)Exercise Group.
For Problems 91–98, factor if possible.
91.
\(h^2-24h+144\)Answer.
\((h-12)^2\)
92.
\(9t^2-30tv+25v^2\)93.
\(x^2+144\)Answer.
Cannot be factored
94.
\(98n^4-8n^2\)95.
\(w^8+12w^4+36\)Answer.
\((w^4+6)^2\)
96.
\(q^6-14q^3+49\)97.
\(3s^4-48\)Answer.
\(3(s^2+4)(s+2)(s-2)\)
