Quick Review Flashcards - Click to flip and test your knowledge!
Question
What is the result of the algebraic expansion of $(a + b)^2$?
Answer
$a^2 + 2ab + b^2$
Question
What is the algebraic expansion of the identity $(a - b)^2$?
Answer
$a^2 - 2ab + b^2$
Question
How is the sum $(a + b)^2 + (a - b)^2$ simplified into a single expression?
Answer
$2(a^2 + b^2)$
Question
What is the simplified form of the difference $(a + b)^2 - (a - b)^2$?
Answer
$4ab$
Question
Given $a \ne 0$, what is the expansion of $(a + \frac{1}{a})^2$?
Answer
$a^2 + \frac{1}{a^2} + 2$
Question
Given $a \ne 0$, what is the expansion of $(a - \frac{1}{a})^2$?
Answer
$a^2 + \frac{1}{a^2} - 2$
Question
How is $a^2 + \frac{1}{a^2}$ expressed in terms of $(a + \frac{1}{a})$?
Answer
$(a + \frac{1}{a})^2 - 2$
Question
How is $a^2 + \frac{1}{a^2}$ expressed in terms of $(a - \frac{1}{a})$?
Answer
$(a - \frac{1}{a})^2 + 2$
Question
What is the result of the sum $(a + \frac{1}{a})^2 + (a - \frac{1}{a})^2$?
Answer
$2(a^2 + \frac{1}{a^2})$
Question
What is the constant result of the difference $(a + \frac{1}{a})^2 - (a - \frac{1}{a})^2$?
Answer
$4$
Question
An equation that is true for all values of its variables is called an _____.
Answer
identity
Question
Formula: $(a + b)^3$
Answer
$a^3 + 3a^2b + 3ab^2 + b^3$
Question
Formula: $(a - b)^3$
Answer
$a^3 - 3a^2b + 3ab^2 - b^3$
Question
How is $(a + b)^3$ expressed as a sum of cubes and a factored product?
Answer
$a^3 + b^3 + 3ab(a + b)$
Question
How is $(a - b)^3$ expressed as a difference of cubes and a factored product?
Answer
$a^3 - b^3 - 3ab(a - b)$
Question
What formula calculates $a^3 + b^3$ using the sum $(a + b)$ and product $ab$?
Answer
$(a + b)^3 - 3ab(a + b)$
Question
What formula calculates $a^3 - b^3$ using the difference $(a - b)$ and product $ab$?
Answer
$(a - b)^3 + 3ab(a - b)$
Question
Expansion: $(a + \frac{1}{a})^3$
Answer
$a^3 + \frac{1}{a^3} + 3(a + \frac{1}{a})$
Question
Expansion: $(a - \frac{1}{a})^3$
Answer
$a^3 - \frac{1}{a^3} - 3(a - \frac{1}{a})$
Question
Formula for $a^3 + \frac{1}{a^3}$ in terms of $(a + \frac{1}{a})$
Answer
$(a + \frac{1}{a})^3 - 3(a + \frac{1}{a})$
Question
Formula for $a^3 - \frac{1}{a^3}$ in terms of $(a - \frac{1}{a})$
Answer
$(a - \frac{1}{a})^3 + 3(a - \frac{1}{a})$
Question
In algebra, if $a + b + c = 0$, what is the value of $a^3 + b^3 + c^3$?
Answer
$3abc$
Question
What is the expansion of the product $(x + a)(x + b)$?
Answer
$x^2 + (a + b)x + ab$
Question
What is the expansion of the product $(x + a)(x - b)$?
Answer
$x^2 + (a - b)x - ab$
Question
What is the expansion of the product $(x - a)(x + b)$?
Answer
$x^2 - (a - b)x - ab$
Question
What is the expansion of the product $(x - a)(x - b)$?
Answer
$x^2 - (a + b)x + ab$
Question
What is the full expansion of the trinomial square $(a + b + c)^2$?
Answer
$a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$
Question
How is $(a + b + c)^2$ expressed using a single factored term for the cross-products?
Answer
$a^2 + b^2 + c^2 + 2(ab + bc + ca)$
Question
What is the expansion of $(a + b - c)^2$?
Answer
$a^2 + b^2 + c^2 + 2ab - 2bc - 2ca$
Question
What is the expansion of $(a - b + c)^2$?
Answer
$a^2 + b^2 + c^2 - 2ab - 2bc + 2ca$
Question
What is the expansion of $(a - b - c)^2$?
Answer
$a^2 + b^2 + c^2 - 2ab + 2bc - 2ca$
Question
Formula: $(x + a)(x + b)(x + c)$
Answer
$x^3 + (a + b + c)x^2 + (ab + bc + ca)x + abc$
Question
Simplify the product $(a + b)(a^2 - ab + b^2)$.
Answer
$a^3 + b^3$
Question
Simplify the product $(a - b)(a^2 + ab + b^2)$.
Answer
$a^3 - b^3$
Question
Identity: $(a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) = \dots$
Answer
$a^3 + b^3 + c^3 - 3abc$
Question
In the expansion of $(a + b + c)^2$, the sum $a^2 + b^2 + c^2$ is always _____ regardless of the signs of $a$, $b$, or $c$.
Answer
positive
Question
To solve for $a + \frac{1}{a}$ given $a^2 - 5a + 1 = 0$, what operation is performed on every term?
Answer
Dividing by $a$
Question
If $x^2 + y^2 + z^2 - xy - yz - zx = 0$, what must be true about the variables $x, y,$ and $z$?
Answer
$x = y = z$
Question
Concept: Difference of Squares application to $(3x - 2y + 4)(3x - 2y - 4)$
Answer
Definition: Group $(3x - 2y)$ as a single term to form $(a+4)(a-4)$, resulting in $(3x - 2y)^2 - 16$.
Question
In the expansion of $(x+a)(x+b)(x+c)$, the coefficient of $x^2$ is the _____ of $a$, $b$, and $c$.
Answer
sum
Question
In the expansion of $(x+a)(x+b)(x+c)$, the constant term is the _____ of $a$, $b$, and $c$.
Answer
product
Question
The formula $(a+b+c)^2 - (a^2+b^2+c^2)$ is used to find the value of which expression?
Answer
$2(ab + bc + ca)$
Question
What is the relationship between $(a - b)^2$ and $(a + b)^2$ involving $ab$?
Answer
$(a - b)^2 = (a + b)^2 - 4ab$
Question
What is the relationship between $(a + b)^2$ and $(a - b)^2$ involving $ab$?
Answer
$(a + b)^2 = (a - b)^2 + 4ab$
Question
If $a + \frac{1}{a} = m$, then $a^2 + \frac{1}{a^2} = \dots$
Answer
$m^2 - 2$
Question
If $a - \frac{1}{a} = n$, then $a^2 + \frac{1}{a^2} = \dots$
Answer
$n^2 + 2$
Question
In the expansion of $(x + a)(x + b)$, the coefficient of $x$ is the _____ of $a$ and $b$.
Answer
sum
Question
What value results from $(a + \frac{1}{a})^2 - (a^2 + \frac{1}{a^2})$?
Answer
$2$
Question
If $a^2 + b^2 + c^2 - ab - bc - ca$ is written as a sum of squares, the first term is $\frac{1}{2}(a - b)^2$. What is the second term?
Answer
$\frac{1}{2}(b - c)^2$
Question
When expanding $(a - b)^3$, the term $3ab^2$ has a _____ sign.
Answer
positive
Question
When expanding $(a - b)^3$, the term $b^3$ has a _____ sign.
Answer
negative
Question
In the expression $x^2 + y^2 + z^2 - xy - yz - zx$, if all variables are different, the value is always _____.
Answer
positive
Question
What is the coefficient of $x$ in the expansion of $(x + 8)(x - 10)$?
Answer
$-2$
Question
What is the constant term in the expansion of $(x - 8)(x + 10)$?
Answer
$-80$
Question
If $a+b+c=l$, $ab+bc+ca=m$, and $a^2+b^2+c^2=n$, what identity links $l$, $m$, and $n$?
Answer
$l^2 = n + 2m$
Question
The expression $(x + y + z)(x + y - z)$ can be viewed as a difference of squares by grouping _____.
Answer
$(x + y)$
Question
To calculate the area of a rectangle using its diagonal ($d$) and perimeter ($P$), you first find the sum of length and breadth ($s = \frac{P}{2}$), then the area is $\frac{s^2 - d^2}{2}$. What expansion identity justifies this?
Answer
$(x+y)^2 = x^2 + y^2 + 2xy$
Question
Given $x + \frac{1}{x} = 2$, the value of $x^n + \frac{1}{x^n}$ for any positive integer $n$ is _____.
Answer
$2$
Question
If $a/b = b/c$, then $ac = b^2$. Using this, $(a+b+c)(a-b+c)$ simplifies to _____.
Answer
$a^2 + b^2 + c^2$
Question
The formula $a^3 + b^3 + c^3 - 3abc$ equals zero if $a+b+c=0$ or if _____.
Answer
$a=b=c$