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SAT Math Essentials > Monomials, Polynomials, Linear and Quadratic Equations, System of Equations > Flashcards

Monomials, Polynomials, Linear and Quadratic Equations, System of Equations Flashcards

1
Q

What is the definition of an equality?

A

An equality is a mathematical statement that two expressions are equal.

The value of the left side equates to the value of the right side.

Example:

3 + 3 = 6

2x + 5x = 7x

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2
Q

What is the definition of an equation?

A

An equation is an equality in which the unknown has a specific value.

Example:

2x = 8

In this case x can only one value, 4. 4 is a solution of the equation above.

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3
Q

What equations do we call equivalent?

A

Equations that have the same solution are called equivalent equations.

In some cases, you may need to transform the original equation into an equivalent to solve it.

Example:

2x + 4 = 8 and 2x + 4 - 8 = 0

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4
Q

Define:

  • Variable
  • Constant
  • Coefficient
A
  • A variable is a symbol for an unknown number
  • A coefficient is a multiplicative factor of a variable
  • Constant is a fixed value

Example:

5x - 2 = 8

5 is a coefficient, x is a variable, 2 and 8 are constants.

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5
Q

How do you solve an equation with one variable?

A
  • Simplify each side of the equation if necessary
  • Isolate the variable by using inverse operations

The result will be an equation with a variable on one side and a real number on the other.

Example:

6x - 8 = -3x + 10

Get all the x’s on one side by adding 3x to both sides: 9x - 8 = 10. Then, add 8 to both sides: 9x = 18. Then, divide both sides by 9: x = 2

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6
Q

In order to solve a linear equation, you might need to simplify it. What operations might be necessary?

A

You may need to do all or some of these to simplify the original equation:

  1. Combine similar terms within grouping symbols
  2. Use distributive property
  3. Remove unnecessary parentheses
  4. Combine like terms

Once you’ve simplified the original equation, proceed with isolating the variable by using inverse operations.

Example:

4 (3x + 3 - 8x) - (-6x) = 26

  1. 4 (3 - 5x) - (-6x) = 26
  2. 12 - 20x - (-6x) = 26
  3. 12 - 20x + 6x = 26
  4. 12 - 14x = 26
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7
Q

What are inverse operations?

A

Two operations are inverse to each other when one operation reverses the effect of the other operation.

Addition and subtraction are inverse operations. Multiplication and division are inverse operations.

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8
Q

How would you use inverse operations to solve the following equations?

z + 15 = 32

5x = 65

A

z + 15 = 32

15 is added to z. To find z, subtract 15 from 32.

5x = 65

5 and x are multiplied. To find x, divide 65 by 5.

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9
Q

What does it mean “to isolate the variable” in an equation?

A

Isolating the variable is the process the result of which is an equation with the variable on one side and a real number on the other side.

To isolate the variable, use inverse operations.

Example:

12 - 14x = 26

  • 14x = 26 - 12
  • 14x = 14
  • x* = -1
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10
Q

Formulate the rule of equality for solving equations.

A

In plain language, whatever you do to one side of the equation, you must do to the other.

The rule of equality states that the same operation using equal numbers must be done on both sides of an equation.

  • Example:*
  • x* + 4 = 6
  • x* + 4 + 3 = 6 + 3

If for some reason you need to add 3 to the left side of the equation, you must also add 3 to the right side.

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11
Q

What steps might be necessary to solve an inequality?

A

Solve an inequality the same way you would solve an equation, i.e. simplify and isolate the variable.

*** Note: When multiplying or dividing both side of the inequality by the same negative number, it reverses the direction of the sign of the inequality.

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12
Q

How does multiplying or dividing an inequality by a negative number affect the direction of the sign of the inequality?

A

Multiplying or dividing an inequality by a negative number reverses the direction of the inequality.

Example:

-2x > 10 Dividing both sides by -2

x -5

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13
Q

Define:

A monomial.

A

A monomial is an expression that has one term.

It’s either a real number, a variable, or the product of a real number and one or more variables.

*** A variable cannot be in the denominator.

Example:

5x, 3a2b, 10

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14
Q

How do you figure out the degree of a monomial?

A

The degree of a monomial is the sum of the degrees (exponents) of its variables.

Example:

x5y3z

5 + 3 + 1 = 9

9 is the degree of this monomial.

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15
Q

Define:

A polynomial.

A

A polynomial is the sum or the difference of monomials.

*** A polynomial must have the same variable.

Example:

5x2 + 6x - 7 or

x - 4

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16
Q

What are the terms of a polynomial?

A

Monomials that make up a polynomial are called its terms.

*** Terms are separated by addition or subtraction signs, but never by multiplication signs.

Example:

15 + 10x

15 and 10x are the terms of this polynomial.

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17
Q

How many terms do binomials and trinomials have?

A
  • Binomial is a polynomial with two terms
  • Trinomial is a polynomial with three terms
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18
Q

How do you simplify a polynomial?

A

To simplify a polynomial, combine similar terms.

Similar terms contain the same variables and same exponents.

Example:

3 + 5x2 + 10x + 24x2 - 3x + 7 =

10 + 7x + 29x2

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19
Q

How do you figure out the degree of a polynomial?

A

The degree of a polynomial is the highest degree of any of its terms.

Example:

7x4 + 10x

Evaluate both terms. The exponent of the 1st term is 4; the exponent of the 2nd term is 1. Therefore, the degree of this binomial is 4.

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20
Q

How do you add and subtract monomials?

A

Monomials being added or subtracted must have the same variable(s).

They are called like terms.

Example:

10x + 2x = 12x

10x and 2x are like terms. To add them, just add the coefficients.

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21
Q

How do you multiply monomials?

A

Multiply the coefficients and the variables of the monomials separately. Write the product in the exponential form when multiplying same bases.

Example:

8a x 2a = (8 x 2) (a x a) = 16a2

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22
Q

How do you add or subtract polynomials?

A

Combine the terms with exactly the same variable(s); i.e. like terms.

Example:

4x - 7 + 5x + 3x2 + 3 = 3x2 + 9x - 4

4x and 5x are like terms. So are -7 and 3.

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23
Q

How do you multiply a monomial by a polynomial?

a (b + c + d)

A

To multiply a polynomial by a monomial, use the distributive property.

a(b + c + d) = ab + ac + ad

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24
Q

What acronym helps you remember how to multiply two binomials?

(a + b)(c + d)

A

To multiply two binomials, use the FOIL method.

  • (a + b)(c + d) =*
  • ac + ad + bc + bd*

FOIL is an acronym for:

F - product of the FIRST terms

O - product of the OUTERMOST terms

I - product of the INNERMOST terms

L - product of the LAST terms

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25
Q

How do you multiply polynomials with more than two terms?

(a + b)(c + d + e)

A

To multiply two polynomials with more than two terms, multiply each term of the first polynomial by each term of the second.

Simply use the distributive property, then combine like terms.

(a + b) (c + d + e) =

ac + ad + ae + bc + bd + be

Example:

(x + 3)(x2 + 5x + 6) =

(x + 3)x2 + (x + 3)5x + (x + 3)6 =

  • x3 + 3x2 + 5x2 + 15x* + 6x + 18 =
  • x3 + 8x2 + 21x* + 18
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26
Q

What does the product of the sum and the difference of two monomials equal to?

(a + b)(a - b) = ?

A

(a - b)(a + b) = a2 - b2

The product of the sum and the difference of two monomials equals to the difference of their squares.

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27
Q

(a + b)2 = ?

(a - b)2 = ?

A

(a + b)2 = a2+ 2ab + b2

(a - b)2 = a2- 2ab + b2

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28
Q

(a + b)3= ?

(a - b)3 = ?

A

Below are the formulas for the cube of the sum or the difference of two monomials.

(a + b)3 =

a3 + 3a2b + 3a**b2 + b3

(a - b)3**=

a3 - 3a2b + 3ab2 - b3

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29
Q

Find the greatest common factor (GCF) of the expressions below.

ca + cb = ?

ca - cb = ?

A

Factor out c as shown below:

ca + cb = c(a + b)

ca - cb = c(a - b)

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30
Q

What is a prime polynomial?

A

A prime polynomial is a polynomial that cannot be factored.

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31
Q

Is there a way to factor out the sum of squares of two monomials?

a2 + b2

A

The sum of squares a2+ b2 is a prime polynomial. It cannot be factored out.

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32
Q

How do you factor out the following expression?

a2 - b2

A

Difference of squares formula:

a2 - b2 = (a + b)(a - b)

Example:

36 - 4x2 = 62 - 22x2 = (6 + 2x) (6 - 2x)

33
Q

How do you factor perfect square trinomials?

  • a2 + 2ab + b2*
  • a2 - 2ab + b2*
A

Factor perfect square trinomials as shown below:

a2 + 2ab + b2 = (a + b)(a + b)

= (a + b)2

a2 - 2ab + b2 = (a - b)( a - b)

= (a - b)2

34
Q

How do you factor the sum or the difference of cubes?

  • a3 + b3*
  • a3 - b3*
A

Factor the sum and the difference of cubes as shown below:

  • a3 - b3 = (a - b)(a2 + ab + b2)*
  • a3 + b3 = (a + b)(a2 - ab + b2*

*** Watch the signs. Use SOAP for help memorize it. “Same, Opposite, Always Positive”.

35
Q

What kind of a polynomial is a quadratic equation?

What is the general form of a quadratic equation?

A

A quadratic equation is a polynomial equation of a second degree.

The general form is

ax2 + bx + c = 0 where

  • a* - quadratic coefficient,
  • b* - linear coefficient,
  • c* - constant term,
  • x* - variable.
36
Q

How do you factor a second degree polynomial into prime polynomials?

A
  • Look for GMF (Greatest Monomial Factor)
  • In binomials, look to apply the formula for the difference of squares
  • In trinomials, look for a perfect square or a pair of binomial factors

*** Remember that NOT all polynomials can be factored.

37
Q

At most, how many solutions does a quadratic equation have?

A

At most, any quadratic equation has two solutions. They are also called roots of the equation.

Some equations may have only one solution or no real solutions.

38
Q

What methods do you know for solving a quadratic equation?

A

There are four methods to solving quadratic equations:

  • Factoring
  • Using square root property
  • Completing the square
  • Using quadratic formula

*** The most used methods for the SAT problems are factoring and using square root property.

39
Q

Out of four methods (factoring, completing the square, using the quadratic formula or using the square root property), how do you know which method to use to solve a quadratic equation at hand?

A
  • Factoring can be used if
  • ax2 + bx + c* can be factored
  • Use the square root property method if
  • x2 = c* or x2 + b = c

where c is not zero

  • The quadratic formula and completing the square methods can be used for any quadratic equation
40
Q

How do you use factoring method to solve a quadratic equation?

ax2 + bx + c = 0

A
  • If possible, factor the left side of the equation
  • Make each factor equal to zero and solve two linear equations
  • Example:*
  • x2 + 15 = 8x* ⇒ x2 - 8x + 15 = 0

(x - 3)(x - 5) = 0

  • x* - 3 = 0 or x - 5 = 0
  • x* = 3 or x = 5
41
Q

How do you factor a quadratic equation when the leading coefficient a = 1?

ax2 + bx + c = 0

*** This type of quadratic equations is most commonly used on the SAT. Become an expert in simple factoring.

A

Here is the teorem of the sum and the product of the roots in a quadratic equation:

  • x2 + bx + c* = 0
  • The sum of the roots is the negative of b coefficient:

root 1 + root 2 = -b

  • The product of the roots is the constant term c:

root 1 * root 2 = c

42
Q

Find the roots of the following quadratic equation.

x2 + 5x + 6 = 0

A

x2 + 5x + 6 = 0

The sum of the roots equals -5. The product of the roots equals 6. You need to find two numbers that satisfy both conditions. Since the product has to be positive, the numbers have to be the same sign. But since the sum has to be negative, you know that these numbers are negative.

Root 1 = -2
Root 2 = -3

43
Q

How do you factor the following quadratic equation?

x2 + 5x + 6 = 0

A

x2 + 5x + 6 = 0

The sum of the roots equals -5. The product of the roots equals 6.

Root 1 = -2
Root 2 = -3

Now, let’s factor the equation.

R1 = -2 ⇒ x + 2 = 0
R2 = -3 ⇒ x + 3 = 0

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

44
Q

If both c and b in a quadratic equation are positive, what can you conclude about the roots of this equation?

x2 + bx + c = 0

A

The roots of this equation are both negative.

Given that c is positive, the product of the roots must be positive and therefore, the roots must be the same sign.

root 1 * root 2 = c
(-root 1) * (-root 2) = c

Since the sum of the roots equals to the negative of b, you can conclude the roots of the equation are both negative.

root 1 + root 2 = -b

  • Example:*
  • x2 + 7x* + 12 = 0

root 1 = -3 root 2 = -4

45
Q

If c is positive and b is negative in a quadratic equation, what can you conclude about the roots of this equation?

x2 - bx + c = 0

A

The roots of this equation are both positive.

Given than c is positive, the product of the roots is positive. Therefore, the roots must have the same sign.

root 1 * root 2 = c
( -root 1) * ( -root 2) = c

The sum of the roots equals to the negative of b. Negative of a negative is positive. Two negative numbers cannot add up to a positive sum, therefore, the roots must be positive.

root 1 + root 2 = - (-b) = b

  • Example:*
  • x2 - 7x* + 12 = 0

root 1 = 3 root 2 = 4

46
Q

If c is negative and b is negative in a quadratic equation, what can you conclude about the roots of this equation?

x2 - bx - c = 0

A

The roots are of opposite signs and the larger root is positive.

Given than c is negative, the product of the roots is negative. Therefore, the roots must be of opposite signs.

root 1 * ( -root 2) = -c
(-root 1) * root 2 = -c

The sum of the roots equals to the negative of b. Negative of a negative is positive. The larger root of the equation must be positive.

root 1 + root 2 = - (-b) = b

  • Example:*
  • x2 - 5x* - 6 = 0

root 1 = 6 root 2 = - 1

47
Q

If c is negative and b is positive in a quadratic equation, what can you conclude about the roots of this equation?

x2 + bx - c = 0

A

The roots of this equation are of opposite signs and the larger root is negative.

Given than c is negative, the product of the roots is negative. Therefore, the roots must be of opposite signs.

root 1 * ( -root 2) = -c
(-root 1) * root 2 = -c

The sum of the roots equals to the negative of b. The larger root of the equation must be negative.

root 1 + root 2 = -b

  • Example:*
  • x2 + 5x* - 6 = 0

root 1 = -6 root 2 = 1

48
Q

How do you factor a quadratic equation when the leading coefficient a is not 1?

  • ax2 + bx + c*
  • a, b, c* are real numbers and a is not zero and not 1.
A

The theorem states that:

  • The sum of the roots equals the negative of the quotient of b and a:

root1 + root 2 = - b/a

  • The product of the roots equals the quotient of c and a:

root1 * root2 = c/a

49
Q

Find the roots of the following quadratic equation?

2x2 + x - 6 = 0

A

2x2 + x - 6 = 0

The sum of the roots equals the negative of the quotient of b and a:

root1 + root 2 = - b/a = -1/2

The product of the roots equals the quotient of c and a:

root1 * root2 = c/a = -6/2 = -3

The numbers that satisfy both conditions above are -2 and 3/2.

50
Q

How do you use the square root method to solve a quadratic equation?

A

Isolate the squared variable on one side and have the constant on the other side. Then, take the square root of each side of the equation.

*** Remember, the square root of a number yields both a positive and a negative value.

Example:

3x2 - 9 = 3 ⇒ 3x2 = 12

x2 = 4 ⇒ x = 2 or -2

51
Q

When should you use the square root method for solving a quadratic equation?

A

The square root method is best to use when the equation only contains x2 term.

  • ax<span>2</span> = c*
  • Example:*

(x - 4)2 = 9 - take the square root of each side of the equation.

  • x* - 4 = 3 ⇒ x = 7
  • x* - 4 = -3 ⇒ x = 1
52
Q

Write the quadratic formula.

A

<dl>
<dd>
<span>You can find solutions to any <em>ax2 + bx + c</em> quadratic equation by using the <strong>quadratic<em> </em>formula</strong>:</span>
</dd>
<dd>
</dd>
<dd>
<img></img>
</dd>
</dl>

53
Q

When do you use the quadratic formula to solve a quadratic equation?

A

<dl>
<dd>
<span>You can use the quadratic formula on any and all quadratics in the form of </span>
</dd>
<dd>
<em><span>ax2 + bx + c </span></em>
</dd>
<dd>
<span>where <em>a</em> is not zero</span><em><span>. </span></em>
</dd>
<dd>
</dd>
<dd>
</dd>
<dd>
<img></img>
</dd>
<dd>
</dd>
<dd>
<span><span>Plug in the values of <em>a, b</em> and <em>c</em> into the formula. </span></span>
</dd>
</dl>

54
Q

Write the formula used to calculate the discriminant in a quadratic equation.

A

In a quadratic equation in the form of ax2 + bx + c the discriminant is calculated by using the formula below:

D = b2 - 4ac

55
Q

How does the discriminant help determine the solutions for a quadratic equation?

A

D = b2 - 4ac

  • If D > 0, the equation has two real roots
  • If D = 0, the equation has one real root
  • If D no real roots

The discriminant determines the number and the kinds of solutions to a quadratic equation in the form of ax2 + bx + c.

56
Q

How do you use completing the square method to solve a quadratic equation?

A

ax2 + bx + c = 0

  • Transpose c to the right side
  • Add a square of half of b coefficient to both sides to make the left side a perfect square

Example:

x2 + 6x + 2 = 0 - transpose
x2 + 6x = -2 - add the square of half of 6
x2 + 6x + 9 = -2 + 9
(x + 3)2 = 7 - use the square root method
x = −3 ±

57
Q

When do you use completing the square method to solve a quadratic equation?

A

ax2 + bx + c = 0

This method works when a equals 1. If a is not 1, divide all terms of the equation by a.

*** This method is not commonly used to solve equations on the SAT. You will either factor a quadratic or use the quadratic formula to find the roots on the test.

58
Q

What do we call a system of linear equations?

A

When you have more than one linear equation with the same set of unknowns, it’s called a system of linear equations.

The solutions must satisfy every equation in the system.

Example:

2x + 5y = 10
-5y + 3x = 15

59
Q

What methods of solving a system of linear equations do you know?

A

You can solve a system of linear equations using one of these methods:

  • graphing
  • substitution
  • addition or subtraction

The simplest kind of a linear system involves two equations and two variables.

60
Q

How do you solve a system of linear equations with two variables using the graphing method?

A

Simply draw the graph of each equation on the same coordinate plane.

  • If the lines intersect, the coordinates of the point of the intersection are the solution
  • If the lines are parallel, there is no solution
  • If the lines are identical (coinside), each point on the line is a solution
61
Q

How do you solve a system of two linear equations with two variables using the substitution method?

A
  • Isolate the variable with the coefficient 1 in one equation
  • Substitute the same variable with the resulted expression in the equation you haven’t used. Solve the equation
  • Use the solution to substitute the unknown in the first equation and solve

Example:

x + 2y = 18
2x + 7y = 6

x = 18 - 2y - isolate the variable
2(18 - 2y) + 7y = 6 - substitute

x = 18 - 2y
36 - 4y + 7y = 6 ⇒ 3y = -30 ⇒ y = -10

x + 2 (-10) = 18 ⇒ x = 38 - plug in y

62
Q

When should you use the substitution method to solve a system of linear equations?

A

Use the substitution method when the coefficient of one of the variables equals 1 or -1.

Example:

x + 5y = 12
3x - 2y = 15

x = 12 - 5y
3(12 - 5y) - 2y = 15

Solve the second equation for y. Then, plug in y value into the first equation and solve for x.

63
Q

How do you solve a system of two linear equations with two variables using the addition or subtraction method?

A
  • Add or subtract equations to eliminate one variable
  • Solve the equation resulting from it
  • Use the solution to substitute the unknown in either equation and solve

Example:

5x + 3y = 25
x - 3y = 11

Add the equations to eliminate y variable.

6x = 36 ⇒ x = 6 - plug in the x value and solve for y.

64
Q

When should you use the addition or subtraction method in a system of linear equations?

A

Use this method if the coefficients of one of the variables have the same absolute value.

*** To eliminate the variable, subtract one equation from the other if the coefficients are the same. Add the equations if the coefficients are opposite.

65
Q

Sometimes extra steps are necessary before you can use the addition or subtraction method to solve a system of linear equations.

A

Sometimes you need to rearrange the terms in the equations prior to solving them.

Example:

2(x - 3) = -6y + 5
5(y + 2) = x - 3

2x - 6 = -6y + 5
5y + 10 = x - 3

2x + 6y = 11
5y - x = -13

Now you can solve the system of equations above by using the substitution method.

66
Q

You may need to use multiplication with the addition or subtraction method to solve a system of linear equations.

A

You may need to multiply the equations to make the coefficients the same absolute value.

Example:

Evaluate the GCM of the like terms’ coefficients. It’s easier to work with 6 than with 20. Multiply the first equation by 2. Multiply the second equation by 3. Then, use the subtraction method.

3x - 5y = 19
2x - 4y = 16

6x - 10y = 38
6x - 12y = 48

2y = -10 ⇒ y = -5
2x - 4 (-5) = 16 ⇒ x = -2

67
Q

When should you multiply equations prior to using the addition or subtraction method to solve a system of linear equations?

A

Multiply if:

  • The coefficient of a variable is a factor of the coefficient of the same variable in another equation
  • The coefficients are relatively prime or fractional

Multiply one or both equations so that the coefficients of one of the variable will have the same absolute value.

68
Q

How many real numbers satisfy this equation?

2x2 + 4 = 0

A

There are no real numbers that satisfy the equation 2x2 + 4 = 0.

2x2 = -4

x2 = -2

No square root can be taken of a negative number within a system of real numbers.

69
Q

The squares of two integers are equal. The product of these integers could be

(a) -3,000
(b) 5,000
(c) 6,000
(d) -9,000

A

(d) -9,000

If the squares of two integers are the same, the numbers could be opposite like 30 and -30.

70
Q

The product of the roots of

(x - 15)(x + 11) = 0 is

(a) - 165
(b) 26
(c) 155
(d) 165

A

(a) -165

The roots of this equation are 15 and -11. The product must be a negative number.

71
Q

If a + b = 5 and a - b = 15, then a2 - b2 equals

(a) 10
(b) 20
(c) 55
(d) 75

A

(d) 75

The difference of squares formula states that

a2 - b2 = (a + b)(a - b) =

5 x 15 = 75

72
Q

(a + 2) divided by (4 - a2) equals

(a) a + 2
(b) 2 - a
(c) a - 2
(d) 2a

A

(b) 2 - a

Use the difference of squares formula to factor 4 - a2.

a2 - b2 = (a + b)(a - b)

22 - a2 = (2 + a)(2 - a)

Then, reduce the resulting algebraic fraction by eliminating common factors.

73
Q

How do you evaluate an algebraic expression?

A

If you are given the values of the variables, you can evaluate an algebraic expression by plugging those values in and calculating the result.

Example:

Evaluate the algebraic expression

6 - 3x + 28 when x = -3.

Combine the like terms 6 and 28 and plug in (-3) into the expression (watch the signs).

34 - 3 (-3) = 34 + 9 = 43

74
Q

When you asked to express one variable in terms of other variables, what do you do?

A

To “solve or express in terms” means to isolate one variable on one side of the equation, leaving the other variables on the other side of the equation.

Example:

75
Q

Changing words into algebraic expressions is necessary to solve word problems.

What are some of the words you can name that mean the addition operation?

A

Here are some words that translate into addition:

  • combine
  • gain
  • exceeds
  • grow
  • rise
  • in all
  • increased by
  • greater than
  • more than
  • larger than
  • sum
  • total
76
Q

Changing words into algebraic expressions is necessary to solve word problems.

What are some of the words you can name that mean the subtraction operation?

A

Here are some words that translate into subtraction:

  • remove
  • deduct
  • depriciate
  • shorten
  • drop
  • lose
  • lower
  • left
  • difference
  • decreased by
  • fewer
  • less than
  • smaller than
  • take away
77
Q

Changing words into algebraic expressions is necessary to solve word problems.

What are some of the words you can name that mean multiplication operation?

A

Here are some words that translate into multiplication:

  • double
  • twice
  • triple
  • quadruple
  • times
  • squared
  • cubed
  • factor
  • product
  • multiple of
78
Q

Changing words into algebraic expressions is necessary to solve word problems.

What are some of the words you can name that mean division operation?

A

Here are some words that translate into division:

  • ratio
  • average
  • split
  • half
  • per
  • quotient
  • third
  • fourth
  • part of

SAT Math Essentials (11 decks)

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