Exploring Algebraic Identities

Factorize the expression $a^2 + b^2 + 2ab - c^2$.

The expansion of $(x^2 + y^2)^2$ is:

If $x - \frac{1}{x} = 5$, find the value of $x^2 + \frac{1}{x^2}$.

If $x + \frac{1}{x} = 4$, find the value of $x^2 + \frac{1}{x^2}$.

To geometrically model the algebraic identity $(x + a)(x + b) = x^2 + (a + b)x + ab$, a student uses four paper cut-outs that fit together to form a single large rectangle. If the areas of three of these cut-outs are $x^2$, $7x$, and $5x$ square units, what is the area of the fourth cut-out?

What is the simplified value of $(x - y)^3 + (y - z)^3 + (z - x)^3$?

If $3x + 2y = 12$ and $xy = 6$, find the value of $27x^3 + 8y^3$.

A square of side length $(a + b)$ contains a smaller square of side length $(a - b)$ positioned centrally inside it ($a > b > 0$), such that their sides are parallel. The region between the boundaries of the two squares is partitioned into four congruent trapezoids by connecting each vertex of the inner square to the corresponding vertex of the outer square. Which of the following expressions represents the area of exactly one of these trapezoids?

If $a + b + c = 0$, find the value of $\frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b}$.

If $x = \frac{\sqrt{3} + 1}{2}$, find the value of $4x^3 + 2x^2 - 8x + 7$.

What is the zero of the polynomial $p(x) = 2x - 7$?

If $p(x) = 12x$, what is $p(0)$?

Find the zero of $p(x) = \frac{1}{4}x + 2$.

The zero of the polynomial $p(x) = x - 1000$ is:

Factorise $x^3 + 8y^3 + 6x^2y + 12xy^2$.

Factorise $15x^2 - 26x + 8$.

Factorise $x^2 - y^2 + 6y - 9$.

If $f(x) = x^4 + x^3 - 2x^2 + x + 1$ and $g(x) = x - 1$, then $f(\text{zero of } g(x))$ is:

Given $p(x) = x^2 - kx + 4$. If one zero is double the other, find $k$.

If $x = -2$ is a zero of $p(x) = x^3 + 3x^2 + 4x + k$, then $p(1)$ is:

In an area model, if we have $(x+2)$ as the height and $(x+4)$ as the width, what is the area of the bottom-right section (the unit tiles section)?

What are the dimensions of a rectangle with area $x^2 + 13x + 30$?

If an area model for $x^2 + bx + 16$ is a square, what is the value of $b$?

Using the area model for $(x+a)(x+b)$, if $a=3$ and $b=3$, how many unit tiles are there?

An area model for $x^{2} + 25x + 100$ is built. What is the length of the rectangle if its width is $(x+5)$?

How many $x^{2}$ tiles are needed to represent the expression $(2x+1)(3x+2)$?

What is the missing value to make $x^{2} + 14x + \square$ a perfect square model?

To represent $x^2 - 5x - 14$ as a rectangle, you must add a certain number of zero pairs. What is the total number of $x$ tiles (both positive and negative) in the final rectangular arrangement?

How many $x^2$ tiles are needed to model $(2x+3)(3x+4)$?

If you are factoring $3x^2 + 13x - 10$, and you arrange the $x$ tiles into groups of 15 and -2 to form the rectangle, what is one of the resulting dimensions?

What is the value of $(99)^3$ using identity $(100-1)^3$?

If $x + y = 10$ and $x^2 + y^2 = 58$, find $xy$.

Find the middle term in the expansion of $(x+6)^2$.

Simplify $(2x+5)(2x-5)$.

If $a, b,$ and $c$ are distinct real numbers, simplify the following expression using algebraic identities: $\frac{(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3}{(a - b)^3 + (b - c)^3 + (c - a)^3}$

Factorize $125x^3 + \frac{1}{8}$.

If $x + \frac{1}{x} = 3$, find the value of $x^4 + \frac{1}{x^4}$.

If $s = a+b+c$, evaluate $(s-a)^3 + (s-b)^3 + 3(s-a)(s-b)c$.

If $x+y=10$ and $x^2+y^2=68$, find $x^4+y^4$.

If $x^2 - 1$ is a factor of $ax^4 + bx^3 + cx^2 + dx + e$, which of the following is true?

Simplify: $\frac{144 - a^2}{12 - a}$

Simplify the expression: $\frac{x^2 + 13x + 42}{x + 6}$

What is the simplified form of $\frac{x^2 - 100}{x - 10}$?

Simplify: $\frac{x^2 + 12x + 35}{x + 5}$

Simplify the rational algebraic expression: $\frac{x^2 - 2x - 35}{x - 7}$

Simplify the expression: $\frac{3x^2 - 3y^2}{x - y}$

Simplify the expression: $\frac{x^2 - 13x + 40}{x - 8}$

Simplify the rational expression: $\frac{x^4 - y^4}{x^2 - y^2} \cdot \frac{x^2 - xy + y^2}{x^3 + y^3}$

Simplify: $\frac{(x-y)^2 + 2xy}{x^2 + y^2}$

Simplify the expression: $\frac{x^2 + 2x + 1}{x^2 - 1} \div \frac{x+1}{x-1}$

Expand $(4x - 1)^2$.

Identify the coefficient of $ab$ in the expansion of $(a + b)^2$.

What is the result of $(x - y)^2$ when $x=5$ and $y=3$?

Calculate $95 \times 95$ using $(100 - 5)^2$.

Factorize $x^2 - y^2 + 2yz - z^2$.

Simplify the expression $(x - y)^2 - (x + y)^2$.

If $(x + a)(x + b) = x^2 + 5x + 6$, find the value of $a^2 + b^2$.

If $x = 99$, find the value of $x^3 + 3x^2 + 3x + 1$.

Simplify $(x+y)^3 - (x-y)^3 - 6y(x^2-y^2)$.

Find the value of $(28)^3 + (-15)^3 + (-13)^3$.

Which of the following is the correct geometric representation of $(a+b)^2$?

What is the square of $(x - 1)$?

Expand $(x + 1)(x + 9)$.

A rectangle is formed by rearranging the parts of a square side $a$ after removing a square side $b$. What are the dimensions of this new rectangle?

To represent the identity $(x+a)(x+b)$ as a rectangle, we sum the areas of four parts. Which part represents the area $ax$?

In the expansion of $(x+a)^2$, if the total area is $x^2 + 14x + 49$, what is the length of the side $a$ in the geometric model?

If a square with side $(x+y)$ is used to visualize $(x+y)^2$, and the two rectangles each have an area of 15, while $x^2 = 9$, what is the value of $y^2$?

In the identity $(a+b+c)^2$, if $a, b, c$ are the sides of a triangle, what can be said about the sum of the areas of the squares $a^2, b^2, c^2$ relative to the sum of the areas of the rectangles $2ab, 2bc, 2ca$?

If the volume $(a+b)^3$ is $64$ and $a=1$, find the volume of the cuboid $3a^2b$.

In the visualization of $(a-b)^2 = a^2 - 2ab + b^2$, if $a=5$ and $b=2$, what is the area of the resulting square?

Factorise the expression: $x^2 + 14x + 49$.

Factorise the expression: $1 - 27x^3$.

Expand the expression: $(x + 2y + 4z)^2$.

Factorise the expression: $x^2 - x - 20$.

Factorize $x^{2} + 4y^{2} + 4y - x^{2}y^{2} - 1$ is not a standard identity, but consider $x^{2} - x^{2}y^{2} + 4y^{2} - 1 + 4y$. Let's try: Factorize $x^{2} + 10x + 25 - y^{2}$.

Factorize $3x^{2} - 243$

Factorize $p^{3}(q-r)^{3} + q^{3}(r-p)^{3} + r^{3}(p-q)^{3}$

If $x+y+z=0$, what are the factors of $x^3+y^3+z^3$?

Factorize: $x^4 + y^4 + (x+y)^4$

Factorize: $a^3 - b^3 + 1 + 3ab$

If you use 1 $x^2$ tile and 6 $x$ tiles to form a square, how many unit tiles do you need to complete it?

Which dimensions would produce exactly 20 unit tiles in an algebra tile model?

How many total tiles are used to represent $x^2 + 2x + 1$?

To represent $(x+3)^2$, how many $x^2$ tiles, $x$ tiles, and unit tiles are needed respectively?

If an algebra tile arrangement for a polynomial has 1 $x^2$ tile, 11 $x$ tiles, and 28 unit tiles, what are its factors?

How many unit tiles are in a model representing $(x+9)(x+2)$?

To represent $3x^2 + 8x + 4$ using tiles, how many $x^2$ tiles are needed?

A student forms a rectangle with $x^2 + 5x + k$ using algebra tiles. If the rectangle has a width of $(x+2)$, what must the value of $k$ be for the rectangle to be perfectly completed?

If you have $1$ $x^2$ tile and $10$ positive $x$-tiles, what is the maximum number of unit tiles you can have to still be able to form a factorable rectangle where all coefficients are integers?