Simplify Grade 10 Algebraic Expressions

Recent questions

• What is a common denominator in fractions?

  A common denominator is a shared multiple of the denominators of two or more fractions. It allows for the addition or subtraction of fractions by providing a uniform base for comparison. To find a common denominator, one typically identifies the least common multiple (LCM) of the denominators involved. This process ensures that each fraction can be expressed with the same denominator, making it possible to combine them accurately. For example, if you have the fractions \( \frac{1}{3} \) and \( \frac{1}{5} \), the common denominator would be 15, as it is the smallest number that both 3 and 5 can divide into without leaving a remainder.

• How do you factor a trinomial?

  Factoring a trinomial involves rewriting it as a product of two binomials. The process typically starts by identifying the coefficients of the trinomial, which is usually in the form \( ax^2 + bx + c \). To factor it, one must find two numbers that multiply to \( ac \) (the product of the coefficient of \( x^2 \) and the constant term) and add up to \( b \) (the coefficient of \( x \)). For instance, in the trinomial \( x^2 - x - 6 \), the numbers 2 and -3 multiply to -6 and add to -1, leading to the factorization \( (x + 2)(x - 3) \). This method simplifies solving quadratic equations and helps in graphing quadratic functions.

• What is the process to combine fractions?

  Combining fractions involves several steps to ensure that the fractions can be added or subtracted correctly. First, one must find a common denominator, which is typically the least common multiple of the denominators involved. Once the common denominator is established, each fraction is adjusted to have this denominator by multiplying both the numerator and denominator by the necessary factors. After adjusting, the numerators can be combined, either by addition or subtraction, depending on the operation. Finally, the resulting fraction may need to be simplified by factoring the numerator and canceling any common factors with the denominator. This systematic approach ensures accuracy in combining fractions.

• How do you simplify a fraction?

  Simplifying a fraction involves reducing it to its lowest terms, which means that the numerator and denominator share no common factors other than 1. To simplify a fraction, one can start by factoring both the numerator and the denominator to identify any common factors. For example, if you have the fraction \( \frac{-2x + 2}{x - 3} \), you can factor the numerator to get \( \frac{-2(x - 1)}{x - 3} \). If there are common factors in the numerator and denominator, they can be canceled out, leading to a simpler form of the fraction. This process not only makes the fraction easier to work with but also provides a clearer representation of the value it represents.

• What does it mean to adjust fractions?

  Adjusting fractions refers to the process of modifying the numerators and denominators of fractions so that they share a common denominator, which is essential for performing addition or subtraction. This adjustment typically involves multiplying the numerator and denominator of each fraction by the necessary factors to achieve the common denominator. For instance, if you are working with the fractions \( \frac{1}{3} \) and \( \frac{1}{5} \), and you determine that the common denominator is 15, you would adjust \( \frac{1}{3} \) by multiplying both the numerator and denominator by 5, resulting in \( \frac{5}{15} \), and adjust \( \frac{1}{5} \) by multiplying both by 3, resulting in \( \frac{3}{15} \). This adjustment allows for the fractions to be combined accurately in subsequent calculations.