**Chapter 5, Section 2: Adding and Subtracting
Polynomials **

Def1.** Like (similar) terms** are terms that
have the same variables with the

same exponents.

EX1. Write 3 other terms that are like (similar) to**
2x**^{2}y

EX2. Write 3 other terms that are not like (similar) to **
2x**^{2}y

Note1. To add (subtract) like terms , add (subtract) their
numerical

coefficients and affix their common variable part.

EX3. Add **3xyz** and **-8xyz**

EX4. Subtract **5rs**^{3} from **2rs**^{3}

EX5. Simplify by combining like terms:**5pqr + pqr - 2pqr**

EX6. Simplify by combining like terms :

3a^{2}b - 2ab^{2} + a^{2}b - 5ab^{2} + a^{2}b^{2}

Note2. The distributive property of multiplication over
addition:

a(b + c) = ab + ac

EX7. Simplify: (-2x^{2} + 6x + 5) - (-4x^{2} - 7x + 2)

EX8. Simplify: 3(y^{2} + 2y) - 4(y^{2} - 4) + 2(y - 3)

EX9. Write a polynomial that represents the perimeter of
the quadrilateral

shown below:

EX10. Draw and label a scalene triangle whose perimeter is

5y^{2} - 4y + 6

EX10. Draw an label an isosceles triangle whose perimeter
is

5y^{2} - 4y + 6

EX11. Suppose that you bought a house in 2006 for
$150,000. Because of

the national housing market disaster, this house is depreciating at the

rate of $8000 per year. (A) Write a polynomial function that will

give you the value of this house in x years. (B) Find the value of the

house in 2010.

**Chapter 5, Section 3: Multiplying Polynomials**

Recall:

x^{a}x^{b} = x^{a+b}

(x^{a})^{b} = x^{ab}

Note1: To multiply one monomial by another, multiply the
numerical factors

and multiply the variable factors.

EX1. (3x^{2}yz^{3})(4xy^{4}z^{5})

EX2. (-2ab^{2}c)(-3abc^{4})(-a^{2}b)

Note2. To multiply a polynomials by a monomial, use the
distributive

property and multiply each term of the polynomial by the monomial.

EX3. 3a(4a^{2} + 3a - 4)

EX4. -4r^{2}s(2r^{2}s^{2} - 3rs^{2} + 5rs)

Note 3. To multiply a polynomial by a polynomial, use the
distributive

property repeatedly.

Special Case. To multiply a binomial by a binomial , use
the distributive

property twice. This process is often referred to as F.O.I.L.

The meaning of F.O.I.L. in this process of multiplying two
binomials is

F=the product of the first terms

O=the product of the outer terms

I= the product of the inner terms

L=the product of the last terms

EX5. (3t - 2)(4t + 3)

EX6. (2a - b)(2a +b)

EX7. (x^{3} + 3y^{2})(x^{2}+2y)

EX8. (y + 4)^{2}

EX9. (a + b)^{2}

Note3. The square of a binomial is a
__________________________.

EX10. (2A - B)(4A^{2} + 3AB - B^{2})

EX11.

A) What is the length of each side of the largest square shown in

the figure above?

B) Find the area of the largest square by using its side
length.

C) Find the area of each part of the largest square.

D) Add the areas that you found in part (C).

E) Compare your answers to parts (B) and (D). What should
be

true and why?

**Chapter 5, Section 4: The Greatest Common Factor and
Factoring by**

Grouping

Def. **Prime number**–a natural number that has
exactly two distinct factors

Question: What does it mean to say that “x is a factor of y”?

EX1. Find all the factors of 24.

Find all the factors of 90.

Find the common factors of 24 and 90.

What is the Greatest Common Factor , GCF, of 24 and 90?

What does it mean to say that “x is the GCF of y”?

EX2. Using prime factorization to find the GCF.

Find the prime factorization of 24 and 90.

Note1. The GCF of the two numbers will be a product the
prime number

factors common to the two numbers with each common prime

number factor raised to the lowest power on that factor in either

factorization.

EX3. Use the method of prime factorization to find the GCF
(280 and 294).

EX4. Find the GCF of a^{2}b^{3}c and a^{4}bd

EX5. Find the GCF of 18x^{2}y and 24xy^{3}

Note2. To factor a polynomial, find the GCF of each term
of the

polynomial and use the distributive property: ab + ac = a(b + c)

EX6. Factor 18x^{2}y + 24xy^{3}

EX7. Factor 30r^{2}s^{2}t - 40r^{3}st^{4}

EX8. Factor 14r^{2}s^{3} + 15t^{4}

EX9. Factor 9m^{4}n^{3}p^{2} + 36m^{2}n^{3}p^{4} - 18m^{2}n^{3}p^{5}

EX10. Factor 25t^{4} - 10t^{3} + 5t^{2}

EX11. Factor -18a^{2}b + 12ab^{3}

EX12. Factor 5(a - b) - c(a - b)

Method 2: Factoring by Grouping

EX13. Factor **ac + ad + bc + bd**

EX14. Factor 2c + 2d - cd - d^{2}

EX15. Factor x^{2} - ax - xy + ay

Literal Equations : In this type of problem, you will be
given an equation and

asked to solve for one of the variables in terms of the other

variables.

EX16. Solve r_{1}r_{2} = rr_{1} + rr_{2} for r.

EX17. Solve H(a + b) = 2ab for a.

EX18. Solve x(5y + 3) = y for y.

EX19. Solve AL + GE = BRA for A.