Problem 44 - College Algebra with Corequisite Support

Question

For the following exercises, multiply the polynomials. \((4 m-13)\left(2 m^{2}-7 m+9\right)\)

Step-by-Step Solution

Answer

The product of the polynomials is \(8m^3 - 54m^2 + 127m - 117\).

Step 1: Distribute the first term

Begin by distributing the first term of the first polynomial, which is \(4m\), across each term of the second polynomial \(2m^2 - 7m + 9\). This involves multiplying \(4m\) by each of the terms in the second polynomial.

Step 2: Calculate the first distribution

Calculate \(4m \times 2m^2\), which equals \(8m^3\). Then calculate \(4m \times (-7m)\), which equals \(-28m^2\), and finally \(4m \times 9\), which equals \(36m\).

Step 3: Distribute the second term

Now, distribute the second term of the first polynomial, which is \(-13\), across each term of the second polynomial \(2m^2 - 7m + 9\). This involves multiplying \(-13\) by each of the terms in the second polynomial.

Step 4: Calculate the second distribution

Calculate \(-13 \times 2m^2\), which equals \(-26m^2\). Then calculate \(-13 \times (-7m)\), which equals \(91m\), and finally \(-13 \times 9\), which equals \(-117\).

Step 5: Combine and simplify terms

Combine all the terms from both distributions: \(8m^3\), \(-28m^2\), \(36m\), \(-26m^2\), \(91m\), and \(-117\). Simplify by combining like terms: \(-28m^2\) and \(-26m^2\) combine to \(-54m^2\), and \(36m\) and \(91m\) combine to \(127m\).

Step 6: Write the final polynomial

The simplified polynomial is \(8m^3 - 54m^2 + 127m - 117\).

Key Concepts

Distributive Property

The distributive property is an essential concept in mathematics, especially when it comes to multiplying polynomials. It allows us to break down complex multiplication problems into simpler parts. In the context of polynomials, the distributive property states that you can distribute the multiplication of a term across a sum or difference inside parentheses. For example, when multiplying the polynomials

• \((4m-13)\)
• \((2m^2 - 7m + 9)\),

the first term, \(4m\), is multiplied separately with each term of the second polynomial:

• \(4m \times 2m^2\)
• \(4m \times (-7m)\)
• \(4m \times 9\)

This breaking down into simpler operations makes it much easier to manage the multiplication.
After distributing the first term, we proceed with the second term,

• \(-13\)

by repeating the process.

• \(-13 \times 2m^2\)
• \(-13 \times (-7m)\)
• \(-13 \times 9\)

The distributive property simplifies polynomial multiplication by applying this step-by-step method for each term.

Combining Like Terms

Once all terms have been distributed, the next step in polynomial multiplication is combining like terms. "Like terms" refers to terms that have the same variables raised to the same power. This process helps to reduce the expression into a cleaner, simplified form.
In the problem, after applying the distributive property, we have obtained several terms from both distributions:

• \(8m^3\)
• -\(28m^2\)
• \(36m\)
• -\(26m^2\)
• \(91m\)
• -\(117\)

The like terms within this list can be grouped together:

• Combine \(-28m^2\) and \(-26m^2\) to become \(-54m^2\)
• Combine \(36m\) and \(91m\) to become \(127m\)

Notice how terms with different powers (e.g., \(8m^3\) or constants like \(-117\)) do not combine with each other. This ensures the polynomial remains ordered from the highest to the lowest power.

Polynomial Simplification

Simplifying polynomials is the last major step in polynomial multiplication. It involves writing the polynomial expression in its most reduced form after all like terms have been combined.
In our example, after distributing the terms and combining like terms, we derive the polynomial:

• \(8m^3 - 54m^2 + 127m - 117\)

This expression is now simplified and contains no additional like terms that can be combined. To ensure it's fully simplified, check the following:

• All terms are arranged in decreasing order of their exponent powers.
• There are no unnecessary coefficients (like a 1 next to variables).
• The expression is as concise as possible, with all like terms already combined.

Such simplification not only makes the polynomial easier to read and interpret but also prepares it for further operations or evaluations in more complex mathematical contexts.