Algebra

Question 1

What is meant by a set and an element?

Answer 1

A set is a collection of objects. For example, we can speak of the set of even numbers between 5 and 11, namely, 6, 8, and 10. An object in a set is called an element of that set.

Question 2

What is meant by an algebraic expression?

Answer 2

If numbers, represented by symbols, are combined by any or all of the operations of addition, subtraction, multiplication, division, exponentiation, and extraction of roots, then the resulting expression is called an algebraic expression.§

Question 3

Describe the algebraic expression 5ax^3 - 2bx + 3 in terms of its terms, factors, coefficients and constants

Answer 3

The algebraic expression 5ax3 - 2bx C 3 consists of three terms: C 5ax3 ; -2bx, and C3. Some of the factors of the first term, 5ax , are 5;a;x;x ;x ;5ax, and ax . Also,
33 5a is the coefficient of x , and 5 is the numerical coefficient of ax . If a and b represent fixed numbers throughout a discussion, then a and b are called constants.

Question 4

How does the number of terms affect the name of an expression?

Answer 4

Algebraic expressions with exactly one term are called monomials. Those having exactly two terms are binomials, and those with exactly three terms are trinomials. Algebraic expressions with more than one term are called multinomials. Thus, the
multinomial 2x - 5 is a binomial; the multinomial 3 y C 2y - 4y2 is a trinomial.

Question 5

What is meant by a polynomial?

Answer 5

A polynomial in x is an algebraic expression of the form:

cn x^n + c xn-1 X^n-1 +… C1x xC0

where n is a nonnegative integer and the coefficients c0; c1,….., cn are constants with c ¤ 0.

Question 6

What is meant by the degree of a polynomial?

Answer 6

We call n the degree of the polynomial. So, 4x^3 - 5x^2 C x - 2 is a polynomial in x of degree 3, and y5 - 2 is a polynomial in y of degree 5.

Question 7

What is meant by a nonzero constant?

Answer 7

A nonzero constant is a polynomial of degree zero; thus, 5 is a polynomial of degree zero. The constant 0 is considered to be a polynomial; however, no degree is assigned to it.

Question 8

How would you simplify 3{2x[2x + 3] + 5[4x^2 - (3 - 4x)]} ?

Answer 8

We first eliminate the innermost grouping symbols (the parentheses). Then we repeat the process until all grouping symbols are removed—combining similar terms whenever possible. We have

3{2x[2x + 3] + 5[4x^2 - (3 - 4x)]}
= 3{2x[2x + 3] + 5[4x^2 - 3 -+4x]}
= 3{4x^2 + 6x + 20x^2 - 15 -+20x]}
= 72x^2 +78x -45

Question 9

Multiply (ax + c) (bx + d)

Answer 9

he distributive property is the key tool in multiplying expressions. For example, to multiply ax + c by bx + d we can consider ax + c to be a single number and then use the distributive property:
(ax + c) (bx + d)
= (ax + c)bx + (ax + c)d
= abx^2 +cbx + adx + cd
= abx^2 + (cb + ad)x + cd

Thus, (ax + c) (bx + d) = abx^2 + (cb + ad)x + cd

e.g (2x + 3) (x - 2)
= 2(1)x^2 (3(1) + 2(-2))x + 3(-2)
= 3x^2 - x - 6

Question 10

What is this product called?:

(x+a)^2 = x^2 + 2ax + a^2

Answer 10

Sum of a square

Question 11

Describe the special product ‘square of a difference’

Answer 11

(x - a)^2 = x^2 - 2ax + a^2

Question 12

Describe the special product: product of sum and difference

Answer 12

(x - a) (x+a) = x^2 - a^2

Question 13

Describe the special product: cube of a sum

Answer 13

(x+a)^3 = x^3 + 3ax^2 + 3a^2x + a^3

Question 14

Describe the special product: cube of a difference

Answer 14

(x-a)^3 = x^3 - 3ax^2 + 3a^2x - a^3

Question 15

Describe how you would divide this polynomial by the monomial:
(x^3 + 3x) / x

Answer 15

(x^3 + 3x) / x
= x^3 / 3 + 3x / x
=x^2 + 3

Question 16

How do you divide a polynomial by a polynomial?

Answer 16

To divide a polynomial by a polynomial, we use so-called long division when the degree of the divisor is less than or equal to the degree of the dividend.

Question 17

Describe how you would Divide 2x3 -14x-5 by x-3.

Answer 17

Here 2x3 - 14x - 5 is the dividend and x - 3 is the divisor. To avoid errors, it is best to write the dividend as 2x3 + 0x2 - 14x - 5. Note that the powers of x are in decreasing order.

We divided x (the first term of the divisor) into 2x3 and got 2x2 .
multiplied 2x by x - 3, getting 2x^3 - 6x^2 . After subtracting 2x - 6x from 2x C 0x , we obtained 6x2 and then “brought down” the term -14x. This process is continued until we arrive at 7, the remainder and land on the quotient, 2x2 -6x-4

Question 18

How is the answer to this long division written as?

dividend / divisor = ?

Answer 18

quotient + remainder / divisor

2x2 -6x-4 + 7/x - 3

Question 19

How can you check your answer of the division is correct?

Answer 19

(quotient)(divisor) + remainder = dividend

Question 20

If two or more expressions are multiplied together, the expressions are called ____ of the ____.

Answer 20

If two or more expressions are multiplied together, the expressions are called factors of the product.

Question 21

What is meant by factoring?

Answer 21

The process by which an expression is written as a product of its factors is called factoring.

Question 22

Describe the commo factor rule of factoring

Answer 22

xy + xz = x(y + z)

abx^2 + (cb + ad)x + cd = (ax + c) (bx + d)

Question 23

Describe the perfect square trinomial rule for factoring

Answer 23

x^2 - 2ax + a^2 = (x - a)^2

Question 24

Describe the difference of two squares rule for factoring

Answer 24

x^2 - a^2 = (x - a) (x+a)

Question 25

Describe the sum of two cubes rule of factoring

Answer 25

(x+a)^3 = x^3 + 3ax^2 + 3a^2x + a^3

Question 26

Describe the difference of two cubes rule of factoring

Answer 26

(x-a)^3 = x^3 - 3ax^2 + 3a^2x - a^3

Question 27

What is an equation?

Answer 27

An equation is a statement that two expressions are equal. The two expressions that make up an equation are called its sides. They are separated by the equality sign, =.

Question 28

What can you not let the value of a variable be in an equation?

Answer 28

We never allow a variable in an equation to have a value for which any expression in that equation is undefined. For example, in
y / y-4 = 6
y cannot be 4, because this would make the denominator zero

Question 29

To solve an equation means to find all values of its variables for which the equation is true. These values are called solutions of the equation and are said to satisfy the equation. When only one variable is involved, a solution is also called a ____

Answer 29

root

Question 30

When are two equations said to be equivalent?

Answer 30

Two equations are said to be equivalent if they have exactly the same solutions, which means, precisely, that the solution set of one is equal to the solution set of the other.

Question 31

What three operations guarantee equivalence?

Answer 31

1. Adding (subtracting) the same polynomial to (from) both sides of an equation, where the polynomial is in the same variable as that occurring in the equation.
2. Multiplying (dividing) both sides of an equation by the same nonzero constant.
3. Replacing either side of an equation by an equal expression.

Question 32

What four common operations may not produce equivalent equations?

Answer 32

4. Multiplying both sides of an equation by an expression involving the variable.
5. Dividing both sides of an equation by an expression involving the variable.
6. Raising both sides of an equation to equal powers (indices).

Question 33

What is a linear equation?

Answer 33

A linear equation in the variable x is an equation that is equivalent to one that can be written in the form

ax + b = 0

where a and b are constants and a /= 0

Question 34

What are literal equations and constants?

Answer 34

Equations in which some of the constants are not specified, but are represented by letters, such as a, b, c, or d, are called literal equations, and the letters are called literal constants.

Question 35

l = Prt
l/Pt = Prt / Pt
r = l/Pt

What did we assume by dividing both sides by Pt?

Answer 35

When we divided both sides by Pt, we assumed that Pt /= 0, since we cannot divide by 0.

Question 36

What is meant by a fractional equation?

Answer 36

A fractional equation is an equation in which an unknown is in a denominator.

Question 37

How do you solve a fractional equation?

Answer 37

We first write the equation in a form that is free of fractions. We can do this by multiplying both sides by the common denominator e.g (x-4)(x-3). Then we use standard algebraic techniques to solve the resulting equation.

Question 38

What is the problem in multiplying by this LCD and how do we rectify it?

Answer 38

In the first step, we multiplied each side by an expression involving the variable x. As we mentioned in this section, this means that we are not guaranteed that the last equation is equivalent to the original equation. Thus, we must check whether or not 9 satisfies the original equation.

Question 39

When does an equation have an empty set?

Answer 39

Some equations that are not linear do not have any solutions. In that case, we say that the solution set is the empty set, which we denote by theta.

e.g, you could arrive at a solution of x = -2, however, the original equation is not defined for x D -2 (we cannot divide by zero), so there are no roots. Thus, the solution set is ;. Although -2 is a solution of Equation (3), it is not a solution of the original equation.

Question 40

What is a radical equation?

Answer 40

A radical equation is one in which an unknown occurs in a radicand.

Question 41

How can you solve a radical solution?

Answer 41

To solve this radical equation, we raise both sides to the same power to eliminate the radical. This operation does not guarantee equivalence, so we must check any resulting “solutions.” We begin by isolating the radical on one side. Then we square both sides and solve using standard techniques.

When an equation has two terms involving radicals, first write the equation
so that one radical is on each side, if possible. Then square and solve.

Question 42

What is a quadratic equation?

Answer 42

A quadratic equation in the variable x is an equation that can be written in the form

ax^2 + bx + c = 0

where a, b, and c are constants and a /= 0.

Question 43

What is a useful method of solving quadratic functions?

Answer 43

A useful method of solving quadratic equations is based on factoring e.g:
4x2 - 17x + 15 = 0

Here a=4; b= -17, and c=15. Thus,
x = -b +/- sqrt(b^2 - 4ac) / 2a
x = -(-17) +/- sqrt(-17^2 - 4(4)(15)) / 2(4)
x = 17 +/- sqrt(49) / 8
= (17 +/- 7) / 8

The roots are:
(17 + 7) / 8 = 3
and
(17 - 7) / 8 = 5/4

Question 44

There the equation had two real roots, what other possible outcomes are there

Answer 44

One real root: e.g if the +/- results in the same number

No real roots: e.g if both are complex numbers

Question 45

What is meant by a quadratic form equation? How can uyou solve one?

Answer 45

Sometimes an equation that is not quadratic can be transformed into a quadratic equation by an appropriate substitution. In this case, the given equation is said to have quadratic form e.g

1/x6 + 9/x3 + 8 = 0
= (1/x6)^2 + 9(1/x^3) + 8 = 0

so it is quadratic in 1=x3 and hence has quadratic form. Substituting the variable w for
1=x3 gives a quadratic equation in the variable w, (w2 + 9w + 8 = 0) which we can then solve and substitute for 1/x^3 when we have our roots