Division of Algebraic Expressions Continued

This section discusses the division of polynomials by polynomials and focuses on factorization techniques for algebraic expressions.

The section elaborates on how to divide algebraic expressions, specifically focusing on checking and matching factors when dividing polynomials by polynomials. It includes several examples to demonstrate the process, emphasizing the importance of factorization in simplification.

Division of Algebraic Expressions Continued

This section delves deeper into the concept of dividing algebraic expressions, specifically targeting the division of polynomials by other polynomials. The process begins by factorizing the numerator and the denominator to identify common factors that can be canceled out. The section provides several illustrative examples to clarify how this method is applied effectively.

For instance, to divide the expression (7x² + 14x) by (x + 2), the teacher guides students to first factorize the numerator into 7x(x + 2). This allows for the subsequent simplification by canceling the common factor (x + 2), leading to the result of 7x. Further, examples demonstrate how to handle more complex polynomial divisions, ensuring that students understand the foundational logic behind factorization and division. The exercises challenge students to practice these concepts, reinforcing their learning through application.

Example 16:

Divide $36(x^4 - 2x^3 - 8x^2)$ by $6(x - 4)$.

Solution:

Factoring $36(x^4 - 2x^3 - 8x^2)$, we get: $36(x^4 - 2x^3 - 8x^2) = 36x^2(x^2 - 2x - 8)$
(Expanding and factoring out the common factor)

We can further factor $x^2 - 2x - 8$:
$x^2 - 2x - 8 = (x - 4)(x + 2)$
So,
$36(x^4 - 2x^3 - 8x^2) = 36x^2(x - 4)(x + 2)$
Then, we have:

We cancel the factors $6$ and $x - 4$:

Therefore, $36(x^4 - 2x^3 - 8x^2) = 6(x - 4)$ results in $6x^2(x + 2)$.

Similar Question:

Divide $54(x^4 - 3x^3 - 12x^2)$ by $9(x - 6)$.

Solution:

Factoring $54(x^4 - 3x^3 - 12x^2)$, we get: $54(x^4 - 3x^3 - 12x^2) = 54x^2(x^2 - 3x - 12)$
(Expanding and factoring out the common factor)

We can further factor $x^2 - 3x - 12$:
$x^2 - 3x - 12 = (x - 6)(x + 2)$
Hence, $54(x^4 - 3x^3 - 12x^2)$ simplifies to $54x^2(x - 6)(x + 2)$.
Then, we have:

We cancel the factors $9$ and $x - 6$:

Therefore, $54(x^4 - 3x^3 - 12x^2) = 9(x - 6)$ results in $6x^2(x + 2)$.