Problem 13 - Algebra 1

Question

Divide. Divide \(10 b^{3}-8 b^{2}-5 b\) by \(-2 b\)

Step-by-Step Solution

Answer

The result is \(-5b^{2} + 4b + \frac{5}{2}\)

Step 1: Divide the first term

Start by dividing the first term of the polynomial, which is \(10b^{3}\), by \(-2b\). This gives us \(-5b^{2}\).

Step 2: Divide the second term

Next, divide the second term, which is \(-8b^{2}\), by \(-2b\). This gives us \(4b\).

Step 3: Divide the third term

Lastly, divide the third term, \(-5b\), by \(-2b\). This gives us \(2.5\) or \( \frac{5}{2}\).

Key Concepts

Monomial Division

When dividing one polynomial by another, especially with a monomial, the process is quite straightforward. A monomial is a polynomial with a single term. In our exercise,

• the polynomial is made up of multiple terms: \(10b^3 - 8b^2 - 5b\).
• The monomial divisor is \(-2b\).

To divide the polynomial by the monomial, you apply division to each term of the polynomial separately. For each term, divide its coefficient by the coefficient of the monomial, and the variable part by its corresponding variable part. This requires applying the property of exponents: when dividing like bases, you subtract the exponents. For example, dividing \(b^3\) by \(b\) gives you \(b^{3-1} = b^2\). You repeat this for each term in the polynomial to find the complete quotient.

Negative Coefficient

Dealing with negative coefficients can sometimes be tricky, but understanding them can simplify your division process. A coefficient is the number that multiplies a variable in a term. In division, signs play an essential role.When you divide two numbers with like signs, their quotient is positive. Likewise, dividing two numbers with unlike signs results in a negative quotient. In our example, the monomial \(-2b\) has both a negative sign and a coefficient of \(-2\). Therefore:

• Dividing \(10b^3\) by \(-2b\) gives \(-5b^2\) since the signs are opposite.
• Dividing \(-8b^2\) by \(-2b\) gives \(4b\), with identical signs resulting in a positive outcome.
• For \(-5b\) divided by \(-2b\), the negative signs cancel out, and the result is a positive \(2.5\) or \(\frac{5}{2}\).

The relationship of signs and coefficients plays an essential role in obtaining the accurate output in polynomial division.

Exponents

Exponents are a way to denote repeated multiplication of a number by itself. In polynomial division, understanding how to handle exponents is crucial.When dividing terms with the same base, you subtract the exponents. For instance, if you have \(b^m\) divided by \(b^n\), you apply the rule: \[b^m \div b^n = b^{m-n}\]In our exercise:

• The first term \(10b^3\) divided by \(-2b\) leads to \(-5b^{3-1} = -5b^2\)
• The second term \(-8b^2\) divided by \(-2b\) becomes \(4b^{2-1} = 4b\)
• The last term \(-5b\) divided by \(-2b\) simplifies to a constant because \(b^{1-1} = b^0 = 1\)

Exponents simplify division as they follow predictable patterns, allowing us to manipulate polynomials more efficiently.