Simplify (8x^5+36x^4-20x^3)/(4x^2)

This problem is asking for the simplification of a given algebraic expression. It presents a polynomial, which is (8x^5 + 36x^4 - 20x^3), and asks for it to be divided by a monomial, which is (4x^2). The task is to perform polynomial long division or use properties of exponents and distributive laws to simplify the expression by cancelling common factors and reducing the terms to their simplest form. The final answer should be a simplified polynomial expression where each term is divided by 4x^2.

$\frac{8 x^{5} + 36 x^{4} - 20 x^{3}}{4 x^{2}}$

Answer

Expert–verified

Solution:

Step 1: Simplify the numerator.

Step 1.1: Extract the common factor from the numerator.

$8x^5 + 36x^4 - 20x^3 = 4x^3(2x^2 + 9x - 5)$
Rewrite the numerator as $4x^3(2x^2 + 9x - 5)$ over the denominator $4x^2$.

Step 1.2: Factor by grouping.

For the trinomial $ax^2 + bx + c$, find two numbers that multiply to $ac$ and add to $b$.
Here, $ac = 2 \cdot (-5) = -10$ and $b = 9$.
Rewrite $9x$ as $(-1 + 10)x$.
Group and factor: $4x^3[(2x^2 - x) + (10x - 5)]$.
Factor out the GCF from each group: $4x^3[x(2x - 1) + 5(2x - 1)]$.
Factor by grouping: $4x^3(2x - 1)(x + 5)$.

Step 2: Simplify terms.

Step 2.1: Cancel the common factor of $4$.

$\frac{4x^3(2x - 1)(x + 5)}{4x^2} = \frac{x^3(2x - 1)(x + 5)}{x^2}$.

Step 2.2: Simplify terms.

Cancel the common factors of $x^3$ and $x^2$.
$\frac{x^3(2x - 1)(x + 5)}{x^2} = (x(2x - 1))(x + 5)$.

Step 3: Expand using the distributive property (FOIL).

$(2x^2 - x)(x + 5)$.

Step 4: Simplify and combine like terms.

Multiply and combine like terms: $2x^3 + 10x^2 - x^2 - 5x$.
Final result: $2x^3 + 9x^2 - 5x$.

Knowledge Notes:

To simplify the given expression $(8x^5 + 36x^4 - 20x^3)/(4x^2)$, we use several algebraic techniques:

Common Factor Extraction: We factor out the greatest common factor (GCF) from the terms in the numerator.
Factoring by Grouping: This involves rewriting a polynomial into groups that have a common factor, then factoring out the GCF from each group.
Simplifying Fractions: We cancel out common factors in the numerator and the denominator.
Distributive Property: Also known as the FOIL method for binomials, it involves multiplying each term in one parenthesis by each term in the other.
Combining Like Terms: This means adding or subtracting terms that have the same variable raised to the same power.
Power Rule of Exponents: When multiplying like bases, we add the exponents: $a^m \cdot a^n = a^{m+n}$.
Commutative Property of Multiplication: The order in which two numbers are multiplied does not affect the product: $ab = ba$.

Using these techniques, we can simplify algebraic expressions and solve various types of algebra problems.