Advanced math

Question 1

What are factoring quadratic and polynomial expressions questions?

Answer 1

Factoring quadratic and polynomial expressions questions ask you to rewrite polynomials in their equivalent, factored form.

Question 2

How to factor a quadratic expression in the form x^2+bx+c

Answer 2

To factor a quadratic expression in the form [x^2+bx+c]:
Find two numbers with a product equal to [c] and a sum equal to [b].
The two factors of the expression are each the sum of [x] and one of the numbers from Step 1.

Question 3

How do we factor by grouping?

Answer 3

The first step of factoring [ax^2+bx+c] is familiar: we’re looking for two integers with a product equal to [ac] and a sum equal to [b].
Use the two numbers from Step 2 to split [bx] into two [x]-terms.
Group the resulting expression into two pairs of terms: one pair should have an [x^2]-term and an [x]-term, and the other pair should have an [x]-term and a constant term.
Factor out an expression containing [x] from the pair with an [x^2]-term and an [x]-term. Factor out a constant from the pair with an [x]-term and a constant term. These two pairs should now share a binomial factor.
The shared binomial factor is one factor of the quadratic expression. The expression containing [x] and the constant factored out in Step 5 combine to form the other factor of the quadratic expression.

Question 4

What are the factored form for this quadratic expressions?
a^2+2ab+b^2=
a^2-2ab+b^2=
a^2-b^2=

Answer 4

a^2+2ab+b^2=(a+b)^2
a^2-2ab+b^2=(a-b)^2
a^2-b^2=(a+b)(a-b)

Question 5

What are Exponential expressions?
Give an example.

Answer 5

Exponential expressions are algebraic expressions with a coefficient, one or more variables, and one or more exponents. For example, in the expression: 3x^4
3 is the coefficient
x is the base
4 is the exponent

Question 6

What are rational exponents?
What are radicals?

Answer 6

Rational exponents refer to exponents that are/can be represented as fractions:
ex: 1/2 or -2/3 are all considered rational exponents.
Radicals are another way to write rational exponents. For example, x^(1/2) and √x are equivalent

Question 7

What are the rules pertaining to Adding and subtracting & Multiplying and dividing exponential expressions?

Answer 7

1) When adding and subtracting exponential expressions, we’re essentially combining like terms. That means we can only combine exponential expressions with both the same base and the same exponent.
2) When multiplying two exponential expressions with the same base, we keep the base the same, multiply the coefficients, and add the exponents. Similarly, when dividing two exponential expressions with the same base, we keep the base the same and subtract the exponents.

Question 8

What are the rules pertaining to Raising an exponential expression to an exponent and change of base? What about Negative exponents?

Answer 8

1) When raising an exponential expression to an exponent, raise the coefficient of the expression to the exponent, keep the base the same, and multiply the two exponents.
2) A base raised to a negative exponent is equivalent to [1] divided by the base raised to the opposite of the exponent.

Question 9

What are polynomial expressions?
How do we call a polynomial with 1, 2 or 3 terms

Answer 9

A polynomial expression has one or more terms with a coefficient, a variable base, and an exponent.
1 term= monomial
2 terms= binomials
3 terms= trinomials

Question 10

How to multiply two polynomials

Answer 10

1) Distribute the terms.
2) Multiply the distributed terms according to the exponent rules.
3) Group like terms.
4) For each group of like terms, add or subtract the coefficients while keeping both the base and the exponent the same.
5) Write the combined terms in order of decreasing power.

Question 11

What are rational expressions?

Answer 11

Rational expressions look like fractions that have variables in their denominators (and often numerators too). For example, x^2/(x+3)

Question 12

How To multiply & divide two rational expressions?

Answer 12

1) Multiply
Factor any factorable polynomial expressions in the numerators and the denominators.
Cancel any identical factors that appear in both the numerators and the denominators of the expressions.
Multiply the remaining numerators and multiply the remaining denominators.
2) Divide
Dividing two rational expressions is similar to multiplying; just remember that dividing by an expression is equivalent to multiplying by the reciprocal of the same expression.

Question 13

How to add/subtract 2 rational expressions?

Answer 13

To add and subtract two rational expressions:
1) Find a common denominator for the two expressions. In most cases, the product of the two denominators would work.
2) Rewrite the equivalent form of each rational expression using the common denominator.
3) Add or subtract the numerators of the expressions while retaining the common denominator.
4) Combine like terms and write the result.
5) Factor and/or cancel as needed.

Question 14

What is a composite function?
How to evaluate composite functions at a specific input value? Given a table?

Answer 14

A composite function uses the output of one function as the input of another. For example, for f(g(x)).
1) Plug in the input value for the input variable wherever it appears in the inner function.
2) Perform the operations specified by the inner function to calculate the output. This output becomes the input of the outer function.
3) Plug in the result of Step 2 for the input variable wherever it appears in the outer function.
4) Perform the operations specified by the outer function to calculate the final output.

Given a table:
1) Find the output value for the inner function corresponding to the specific input value. This is also the input value of the outer function.
2) Find the output value for the outer function corresponding to the input of the result of Step 1.

Question 15

What are isolating quantities problems? How to isolate a quantity in an equation/formula?

Answer 15

Many real-world scenarios can be described using equations and formulas with multiple variables. For example, the formula for the area, [A], of a rectangle with length l and width [w] is A=l*w. In this form, area ([A]) is isolated: it is alone on one side of the equation.
If we want to find the equivalent equation, but with l isolated, we can divide both sides of the equation by [w], giving us l=A/w
To isolate a quantity in an equation or formula:
Write down the original equation. If needed, translate the word problem or given context into an equation.
Perform the same operation on both sides of the equation to begin isolating the desired quantity.
Repeat Step 2 until the desired quantity is isolated.

Question 16

What are quadratic equations?

Answer 16

A quadratic equation is an equation with a variable to the second power as its highest power term.

Question 17

How to solve quadratic equations without x-terms?(just x^2 as the unknown)

Answer 17

Isolate x^2.
Take the square root of both sides of the equation. Both the positive and negative square roots are solutions.

Question 18

How To solve a factored quadratic equation using the zero product property?

Answer 18

1) Set each factor equal to [0].
2) Solve the equations from Step 1. 3) The solutions to the linear equations are also solutions to the quadratic equation.

Question 19

What is the quadratic formula?
How to solve a quadratic equation using the quadratic formula?

Answer 19

(-b±sqrt(b^2-4ac))/2a
1) Rewrite the equation in the form [ax^2+bx+c=0].
2) Substitute the values of [a], [b], and [c] into the quadratic formula
3) Evaluate [x]

Question 20

What does this part of the quadratic formula tell us: sqrt(b^2-4ac)?

Answer 20

The [b^2-4ac] portion of the quadratic formula is called the discriminant. The value of [b-4ac] tells us the number of unique real solutions the equation has:
If [b^2-4ac>0], then the equation has [2] unique real solutions.
If [b^2-4ac=0], then the equation has [1] unique real solution.
If [b^2-4ac<0], then the equation has no real solution.

Question 21

What are linear and quadratic systems?

Answer 21

Linear and quadratic systems are systems of equations with one linear equation and one quadratic equation.

Question 22

How to solve a linear and quadratic system?

Answer 22

1) Isolate one of the two variables in one of the equations. In most cases, isolating [y] is easier.
2) Substitute the expression that is equal to the isolated variable from Step 1 into the other equation. This should result in a quadratic equation with only one variable.
3) Solve the resulting quadratic equation to find the [x]-value(s) of the solution(s).
4) Substitute the [x]-value(s) into either equation to calculate the corresponding [y]-values.

Question 23

What are radical, rational, and absolute value equations?

Answer 23

Radical equations are equations in which variables appear under radical symbols.
ex: sqrt(2x-1)=x
Rational equations are equations in which variables can be found in the denominators of rational expressions.
ex: 1/(x+1)=2/x
Absolute value equations are equations in which variables appear within vertical bars (II).
Ix+1I=2 is an absolute value equation.

Question 24

How To solve a radical equation?
How To check for extraneous solutions to a radical equation?

Answer 24

-Isolate the radical expression to one side of the equation.
Square both sides the equation.
Rearrange and solve the resulting equation.
-Solve the radical equation as outlined above.
Substitute the solutions into the original equation. A solution is extraneous if it does not satisfy the original equation.

Question 25

How to solve a rational equation?
How to check for extraneous solutions?

Answer 25

-To solve a rational equation:
Rewrite the equation until the variable no longer appears in the denominators of rational expressions.
Rearrange and solve the resulting linear or quadratic equation.
-To check for extraneous solutions to a rational equation:
Solve the rational equation as outlined above.
Substitute the solution(s) into the original equation. A solution is extraneous if it does not satisfy the original equation.

Question 26

How to solve absolute value equations?

Answer 26

When solving absolute value equations, rewrite the equation as two linear equations, then solve each linear equation. Both solutions are solutions to the absolute value equation.

Question 27

What are quadratic and exponential word problems?

Answer 27

Both quadratic functions and exponential functions can be used to model nonlinear relationships in everyday life, such as the height of a falling object or the population change of a city.

Question 28

What are the Common quadratic and exponential word problem scenarios on the SAT?

Answer 28

1) Area of a rectangle
Because the formula for the area of a rectangle, A=l*w, is commonly known, you’re expected to be able to write quadratic equations modeling rectangular areas and solve for length or width when area is given.
2) Height versus time
Calculate the height of the object at a given time
Calculate the time at which the object is at a given height
A common given height is “the ground”, which means a height of [0] units.
3)Population growth and decay / Compounding interest
Functions modeling these topics typically look like f(t)=a(b)^t, where [a] is the initial value, [b] describes change, and [t] is the variable representing time.

Question 29

What are quadratic functions?

Answer 29

In a quadratic function, the output of the function is based on an expression in which the input to second power is the highest power term.

Question 30

How to graph a quadratic function?

Answer 30

-Evaluate the function at several different values of [x].
-Plot the input-output pairs as points in the [xy]-plane.
-Sketch a parabola that passes through the points.

Question 31

What are the features of a parabola?
When does it hit its maximum/minimum?

Answer 31

-All parabolas have a [y]-intercept, a vertex, and open either upward or downward.
-If the parabola opens upward, then the vertex is the lowest point on the parabola.
If the parabola opens downward, then the vertex is the highest point on the parabola.

Question 32

When does the parabola open upward/downward using the standard form ?

Answer 32

the coefficient of the [x^2]-term, [a], tells us whether the parabola opens upward or downward:
If [a>0], then the parabola opens upward.
If [a<0], then the parabola opens downward.

Question 33

How To identify the features of a parabola from a quadratic equation?
To match a parabola with its quadratic equation?

Answer 33

-Remember which equation form displays the relevant features as constants or coefficients.
Rewrite the equation in a more helpful form if necessary.
Identify the constants or coefficients that correspond to the features of interest.
-Determine the features of the parabola.
Identify the features shown in quadratic equation(s).
Select a quadratic equation with the same features as the parabola.
Plug in a point that is not a feature from Step 2 to calculate the coefficient of the [x^2]-term if necessary.

Question 34

What are the 3 different forms of quadratic equations? And what does each one tell us?

Answer 34

1) The standard form of a quadratic equation, y=ax^2+bx+c shows the [y]-intercept of the parabola:
The [y]-intercept of the parabola is located at (0,c).
2) The factored form of a quadratic equation, y=a(x-b)(x-c), shows the [x]-intercept(s) of the parabola
The [x]-intercepts(root, zero) of the parabola are located at (b,0) and (c,0)
3)The vertex form of a quadratic equation, y=a(x-h)^2+k, reveals the vertex of the parabola. The vertex of the parabola is located at (h,k)

Question 35

What are exponential graphs?

Answer 35

In an exponential function, the output of the function is based on an expression in which the input is in the exponent.

Question 36

When are the graph showing an exponential growth or decay?
How to shift the horizontal asymptote? How to shift the y-intercept?

Answer 36

1) If b>1, then the slope of the graph is positive, and the graph shows exponential growth. As x increases, the value of y approaches infinity. As x decreases, the value of y approaches 0.
If 0<b<1, then the slope of the graph is negative, and the graph shows exponential decay. In this case, as x increases, the value of y approaches 0. As x decreases, the value of y approaches infinity.
2) To shift the horizontal asymptote:
For f(x)=b^x, the value of y approaches infinity on one end and the constant 0 on the other.
For f(x)=b^x+d, the value of y approaches infinity on one end and d on the other.
3) To shift the y-intercept:
For f(x)=b^x+d, the y-intercept is 1+d.
For f(x)=ab^x, the y-intercept is a1=a. In this form, a is also called the initial value.
For f(x)=a*b^x+d, the y-intercept is a+d.

Question 37

what is a polynomial function? And rational functions?

Answer 37

-the output of the function is based on a polynomial expression in which the input is raised to the second power or higher.
-The input of a rational function appears in the denominator of an expression.

Question 38

How To determine the zeros of a polynomial function in factored form?

Answer 38

1) Set each factor equal to 0.
2) Solve the equations from Step 1. The solutions to the linear equations are the zeros of the polynomial function.

Question 39

How To write a polynomial function when its zeros are provided?

Answer 39

1) For each given zero, write a linear expression for which, when the zero is substituted into the expression, the value of the expression is 0.
2) Each linear expression from Step 1 is a factor of the polynomial function.
The polynomial function must include all of the factors without any additional unique binomial factors.
ex:
We are given three real roots, -1, 3, 8. This means there are three corresponding factors.
Since -1+1=0, x+1 is a factor.
Since 3-3=0, x-3 is also factor.
Since 8-8=0, x-8 is the last factor.

Question 40

How can we can determine the y-intercept and end behavior of a polynomial graph?

Answer 40

-The y-intercept happens when x=0, and so is equal to the constant term of the polynomial expression.
-End behavior is just another term for what happens to the value of y as x becomes very large in both the positive and negative directions. For the highest power term ax^n:
If a>0, then y ultimately approaches positive infinity as x increases.
If a<0, then y ultimately approaches negative infinity as x increases.
If n is even, then the ends of the graph point in the same direction.
If n is odd, then the ends of the graph point in different directions.

Question 41

What is the polynomial remainder theorem, and what does it tell us?

Answer 41

-The polynomial remainder theorem states that when a polynomial function p(x) is divided by x-a, the remainder of the division is equal to p(a).
If p(a)=0, then (a,0) is an x-intercept, and x-a is a factor of p(x).
If p(a)≠0, then x-a is not a factor of p(x).

Question 42

When is a rational function is undefined?

Answer 42

A rational function is undefined when division by 0 occurs.