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📘intermediate algebra review

key term - (a + b)(a² - ab + b²)

Citation:

Definition

(a + b)(a² - ab + b²) is a special product in algebra, where the expression on the left-hand side can be factored into the expression on the right-hand side. This term is particularly important in the context of factoring special products, as it demonstrates a specific pattern that can be used to simplify and factor polynomial expressions.

5 Must Know Facts For Your Next Test

  1. The expression (a + b)(a² - ab + b²) is a special product known as the difference of two squares.
  2. When factoring this expression, the resulting factors are (a + b) and (a - b).
  3. The difference of two squares formula can be used to factor any expression of the form $a^2 - b^2$.
  4. Factoring special products like (a + b)(a² - ab + b²) is an important skill in algebra, as it allows for simplifying and solving more complex polynomial equations.
  5. The ability to recognize and factor special products can greatly improve one's problem-solving skills in algebra and lead to a deeper understanding of polynomial operations.

Review Questions

  • Explain the process of factoring the expression (a + b)(a² - ab + b²).
    • To factor the expression (a + b)(a² - ab + b²), we can use the difference of two squares formula. First, we identify the two squares within the expression: $a^2$ and $b^2$. Then, we factor the expression as (a + b)(a - b). This works because (a + b)(a - b) = $a^2 - b^2$, which is the same as the original expression $a^2 - ab + b^2$. By factoring the expression in this way, we can simplify the polynomial and express it as a product of two binomials.
  • How can the factorization of (a + b)(a² - ab + b²) be used to solve polynomial equations?
    • The factorization of (a + b)(a² - ab + b²) can be very useful in solving polynomial equations. If we have an equation of the form $(a + b)(a² - ab + b²) = 0$, we can use the fact that the product of two factors is equal to zero if and only if at least one of the factors is equal to zero. This means we can set each factor equal to zero and solve for the values of a and b that satisfy the equation. This approach allows us to break down a higher-degree polynomial equation into simpler, linear equations that can be more easily solved.
  • Explain how the understanding of the difference of two squares formula, as demonstrated in the factorization of (a + b)(a² - ab + b²), can be applied to other polynomial expressions.
    • The understanding of the difference of two squares formula, as seen in the factorization of (a + b)(a² - ab + b²), can be applied to a wide range of polynomial expressions. The general difference of two squares formula is $a^2 - b^2 = (a + b)(a - b)$. This pattern can be recognized and used to factor other polynomial expressions that fit this form, such as $x^4 - y^4$, $9a^2 - 16b^2$, or $(2x)^2 - (3y)^2$. By identifying the difference of two squares structure, you can quickly factor these expressions and simplify more complex polynomial equations. This skill is essential for success in advanced algebraic problem-solving.