Practice Questions

Critique the following statement: 'To factor $x^2 - 5x - 6$, we need two numbers that multiply to -6 and add to -5. These are -2 and -3. So the factors are $(x-2)(x-3)$.' Justify your critique.

Calculate the value of $98^2$ using a suitable algebraic identity.

Explain the fundamental difference between an algebraic equation and an algebraic identity. Provide one example of each.

If $a^2 + b^2 + c^2 = 125$ and $ab+bc+ca = 50$, evaluate all possible values for $(a+b+c)$ and justify your reasoning.

Write the complete algebraic identity for the expansion of $(a+b+c)^2$.

List the two factors of the expression $x^2 - y^2$.

Justify why the equation $(x+3)^2 = x^2 + 9$ is not an algebraic identity. Propose a value for $x$ that demonstrates this.

State the identity that expresses the difference of two cubes, $a^3 - b^3$, as a product of its factors.

State the algebraic identity for the square of a difference, $(a-b)^2$.

Apply the identity for $(a+b+c)^2$ to expand $(3x - 5y - 2z)^2$.

Apply the identity $(a+b)^2 = a^2+2ab+b^2$ to expand $(2p + 7q)^2$.

Apply a suitable identity to factorize $16x^2 - \frac{y^2}{25}$.

If $a-b = 7$ and $ab = 18$, calculate the value of $a^2+b^2$.

Analyze the expression $9a^2 - 30ab + 25b^2$ and factorize it using an appropriate identity.

Calculate the product of $103 \times 104$ using the identity $(x+a)(x+b) = x^2 + (a+b)x + ab$.

If $x + \frac{1}{x} = 6$, calculate the value of $x^2 + \frac{1}{x^2}$.

Solve for the factors of the quadratic expression $p^2 - 10p + 21$ by splitting the middle term.

Summarize the important algebraic identity that relates the expression $x^3+y^3+z^3-3xyz$ to its factors.

Name the identity that gives the product of $(x+a)$ and $(x+b)$.

Explain the geometric visualization of the identity $(a+b)^2 = a^2 + 2ab + b^2$. Describe how a square with side length $(a+b)$ is partitioned to represent each term of the identity.

Recall the full expansion of the identity for $(x+y)^3$.

Describe the step-by-step process of factoring a quadratic expression of the form $x^2+px+q$ by 'splitting the middle term'. Use the expression $x^2+5x+6$ as an example to explain your steps.

Identify the simplified expression for the product $(x+y)(x^2 - xy + y^2)$.

Describe how the identity $(a+b)^2 = a^2+2ab+b^2$ is modified to get the identity for $(a-b)^2$.

If the expression $p^2 + 14p + 49$ is written in the form of the identity $(a+b)^2$, identify the expressions for 'a' and 'b'.

Calculate the value of $97 \times 103$ without direct multiplication by applying an algebraic identity.

The area of a rectangular field is given by the expression $2x^2 + 9x - 5$ square units. If its length is $2x-1$ units, analyze the expression to find its breadth.

Analyze and factorize the expression $64m^3 + 27n^3$ using the identity for the sum of cubes.

Formulate a general rule for factoring the expression $a^{2n} - b^{2n}$ for any positive integer $n$, and justify your formulation.

Create a word problem for which the solution involves factoring the expression $2x^3 - 2x$. The problem should ask for possible dimensions of a cuboid. Then, provide the solution to your created problem.

Create an example of a trinomial of the form $ax^2 + bx + c$ which is a perfect square, where $a$ is not a perfect square itself. Justify your choice.

Design a geometric proof for the identity $(a-b)^2 = a^2 - 2ab + b^2$, assuming $a > b > 0$. Your proof must include a labeled diagram and a justification based on the areas of the components.

If $x + \frac{1}{x} = 5$, create an expression for $x^3 + \frac{1}{x^3}$ in terms of its value and justify your steps using an algebraic identity.

A student claims that if $x+y+z=0$, then $x^3+y^3+z^3 = 0$. Critique this claim and, if it is incorrect, derive the correct relationship.

A student needs to calculate $103 \times 97$ without direct multiplication. They propose using the identity $(a+b)^2$. Critique this choice and propose a more suitable identity, justifying your selection and providing the calculation.

Propose a method to factor the expression $x^4 + 4y^4$. Justify your method by showing the complete factorization.

Justify whether $x^3 - x$ is always divisible by 3 for any positive integer $x$.

Given that $x+y+z = 12$ and $x^2+y^2+z^2 = 70$, apply a suitable identity to calculate the value of $xy+yz+zx$.

If $p = 2-a$, prove that $a^3 + p^3 + 6ap = 8$. Justify each step of your proof.

Simplify the rational expression $\frac{x^2 - 4}{x^2 - x - 6}$, assuming the denominator is not zero.

If $x + y = 10$ and $xy = 16$, analyze the given information to calculate the value of $x^3 + y^3$.

Evaluate $(x-2y)^3 - (x+2y)^3$ and justify the identities used in your simplification.

Explain how to derive the identity for $(a+b+c)^2$ by using the known identity for $(x+y)^2$. Describe the substitution and expansion steps involved.

If both $(x-3)$ and $(x - \frac{1}{3})$ are factors of the polynomial $px^2 + 7x + r$, formulate a proof to show that $p=r$.

List the eight geometric components (cubes and cuboids) that are formed when a large cube of side length $(a+b)$ is partitioned to visualize the identity for $(a+b)^3$.