Simplify (4b^3-6b^2-b)÷4b

The question is asking to perform the operation of division on a polynomial expression. Specifically, you are to simplify the expression (4b^3 - 6b^2 - b) by dividing it by 4b. The process involves dividing each term of the polynomial by the monomial 4b and simplifying the result to its lowest terms by canceling out common factors in numerators and denominators.

$(4 b^{3} - 6 b^{2} - b) \div 4 b$

Answer

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Solution:

Step 1:

Express the division as a fraction: $\frac{4 b^{3} - 6 b^{2} - b}{4 b}$ .

Step 2:

Extract the common factor $b$ from the numerator.

Step 2.1:

Take $b$ out from $4 b^{3}$ : $\frac{b (4 b^{2}) - 6 b^{2} - b}{4 b}$ .

Step 2.2:

Remove $b$ from $- 6 b^{2}$ : $\frac{b (4 b^{2}) + b (- 6 b) - b}{4 b}$ .

Step 2.3:

Extract $b$ from $- b$ : $\frac{b (4 b^{2}) + b (- 6 b) + b (- 1)}{4 b}$ .

Step 2.4:

Combine the terms with $b$ factored out: $\frac{b (4 b^{2} - 6 b) + b (- 1)}{4 b}$ .

Step 2.5:

Final extraction of $b$ : $\frac{b (4 b^{2} - 6 b - 1)}{4 b}$ .

Step 3:

Eliminate the common $b$ factor from the numerator and denominator.

Step 3.1:

Cross out the common $b$ : $\frac{b (4 b^{2} - 6 b - 1)}{4 b}$ .

Step 3.2:

Simplify the expression: $\frac{4 b^{2} - 6 b - 1}{4}$ .

Knowledge Notes:

The problem involves simplifying an algebraic expression that is divided by a monomial. The process requires understanding of several algebraic concepts:

Fraction Notation for Division: Division of algebraic expressions can be represented as a fraction, where the numerator is the dividend and the denominator is the divisor.
Factoring: This is the process of breaking down an expression into its constituent factors. In this case, factoring out $b$ from each term in the numerator simplifies the expression.
Distributive Property: This property is used when factoring out $b$ from each term. It states that $a (b + c) = a b + a c$ .
Cancellation: When the same factor appears in both the numerator and the denominator of a fraction, it can be cancelled out. This is based on the property that $\frac{a}{a} = 1$ for any non-zero $a$ .
Simplification: The final step in the process is to rewrite the expression in its simplest form after cancelling out common factors.

Understanding these concepts is essential to perform algebraic manipulations and simplify expressions effectively.