Question

Factor the sum or difference of cubes.$8t^3-1$

Short answer

The factorized form of the expression $8t^3-1$ is $(2t-1)(4t^2+2t+1)$.

Step-by-step solution

Identify the cubes

Identify the values of $a$ and $b$. For $8t^3-1$, $a=2t$ (since $(2t{)}^3=8t^3$) and $b=1$ (since $1^3=1$).

Apply the formula

Substitute the values of $a$ and $b$ into the formula to factorise difference of cubes, $(a-b)(a^2+ab+b^2)$. This becomes: $(2t-1)[(2t{)}^2+(2t\cdot 1)+1^2]$

Simplify the expression

Simplify the expression $(2t-1)[(2t{)}^2+2t+1]$, resulting in $(2t-1)(4t^2+2t+1)$.

Key concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. Understanding how to work with algebraic expressions is fundamental in algebra, as these expressions represent quantities that can vary, known as variables. For example, in the expression $8t^3-1$, $t$ is the variable, and the numerical coefficients are 8 and -1, representing constants. Students must recognize the structure of algebraic expressions to factor them effectively, especially when dealing with polynomials, which are algebraic expressions that involve sums of powers of variables.

In our exercise, identifying the cubic terms—$8t^3$ and $1$—and understanding that we are dealing with a difference of cubes helps us use specific formulas for factoring that might not be evident at first glance. Recognizing that $8t^3$ is actually $2t$ raised to the third power is a key insight that leads to the correct factorization of the expression.

Factoring Polynomials

Factoring polynomials is breaking down a polynomial into simpler 'factors' that, when multiplied together, yield the original polynomial. One common type of polynomial factoring is the sum or difference of cubes. Recognizing whether a polynomial can be factored as a sum or difference of cubes requires understanding perfect cubes and cube roots.

For the exercise $8t^3-1$, the critical steps involve recognizing that $8t^3$ and $1$ are both cubes—of $2t$ and $1$, respectively. The general approach is to match the given polynomial with the standard formula for the difference of cubes: $(a^3-b^3)$, where $a$ and $b$ are the cube roots of the first and second terms. It's then possible to apply the formula for factoring, which is heavily dependent on pattern recognition and algebraic manipulation.

After applying the difference of cubes formula, $(a-b)(a^2+ab+b^2)$, the polynomial is factored into two simpler expressions that give insight into its roots and may be further analyzed or simplified.

Sum or Difference of Cubes

The sum or difference of cubes formula is a special method used in algebra for factoring expressions that represent either the sum or difference of two perfect cubes. The formulas are $(a^3+b^3)=(a+b)(a^2-ab+b^2)$ and $(a^3-b^3)=(a-b)(a^2+ab+b^2)$, where $a$ and $b$ are the cube roots of the original terms in the expression.

In the context of our exercise, we use the formula for the difference of cubes to factor the expression $8t^3-1$. After identifying $2t$ and $1$ as the cube roots of each term, we substitute them into the formula to obtain $2t-1$ for $a-b$ and $4t^2+2t+1$ for $a^2+ab+b^2$. This results in the factorized form $2t-1)(4t^2+2t+1)$, completing the process.

Understanding this formula is not just about memorizing it, but also about knowing how to apply it to different algebraic expressions. Doing so successfully requires the ability to recognize patterns and perform algebraic manipulations, which are key skills in higher-level mathematics.