Problem 74 - College Algebra with Modeling and Visualization

Question

Multiply the polynomials. $$(2 x-1)\left(3 x^{2}-x+6\right)$$

Step-by-Step Solution

Answer

The product is \(6x^3 - 5x^2 + 13x - 6\).

Step 1: Apply the Distributive Property

To multiply the polynomials \(2x - 1\) and \(3x^2 - x + 6\), distribute each term of the first polynomial across each term of the second. Start by multiplying \(2x\) with each term of the second polynomial. This gives \(2x imes 3x^2, 2x imes (-x),\) and \(2x imes 6\).

Step 2: Perform the Multiplication with First Term

Calculate each of the products from Step 1:- \(2x imes 3x^2 = 6x^3\)- \(2x imes (-x) = -2x^2\)- \(2x imes 6 = 12x\)

Step 3: Multiply the Second Term Using Distribution

Now distribute \(-1\) across the terms in the second polynomial: \(-1 imes 3x^2, -1 imes (-x),\) and \(-1 imes 6\).

Step 4: Perform the Multiplication with Second Term

Calculate each of the products from Step 3:- \(-1 imes 3x^2 = -3x^2\)- \(-1 imes (-x) = x\)- \(-1 imes 6 = -6\)

Step 5: Combine All Products

Add all the products obtained from Steps 2 and 4:Let us list them: \(6x^3\), \(-2x^2\), \(-3x^2\), \(12x\), \(x\), \(-6\).

Step 6: Simplify the Expression

Combine like terms to simplify the expression:- The \(x^3\) term: \(6x^3\)- The \(x^2\) terms: \(-2x^2 - 3x^2 = -5x^2\)- The \(x\) terms: \(12x + x = 13x\)- Constants: \(-6\)The final simplified polynomial is \(6x^3 - 5x^2 + 13x - 6\).

Key Concepts

Distributive Property

The Distributive Property is a fundamental concept in algebra used extensively in polynomial multiplication. It states that for any three numbers or algebraic expressions, such as \(a\), \(b\), and \(c\), the equation \(a(b + c) = ab + ac\) holds true. This property is what allows us to "distribute" a multiplier to each term within a set of parentheses. In the context of multiplying two polynomials, like \((2x-1)(3x^2-x+6)\), we use the Distributive Property twice:

• First, we distribute \(2x\) across each term in the second polynomial, performing multiplications like \(2x \times 3x^2\), \(2x \times (-x)\), and \(2x \times 6\).
• Second, we do the same with \(-1\), multiplying \(-1 \times 3x^2\), \(-1 \times (-x)\), and \(-1 \times 6\).

This double application of distributing allows us to ensure that every term in one polynomial interacts with every term in the second polynomial, which is crucial in fully expanding the expression.

Combining Like Terms

Once each term from one polynomial has been multiplied by each term of another polynomial through the Distributive Property, the next critical step is combining like terms. Like terms are terms in a polynomial that have the same variables raised to the same powers. In the expression formed from our exercise, you might find several terms with the same degree. For example:

• In our expanded polynomial, \(6x^3\), \(-2x^2\), \(-3x^2\), \(12x\), \(x\), and \(-6\), the like terms are \(-2x^2\) and \(-3x^2\), both having the power of 2 for the variable \(x\). Combining them results in \(-5x^2\).
• Similarly with the terms \(12x\) and \(x\), both are linear terms, \(12x + x = 13x\).

Combining like terms simplifies the algebraic expression, making it easier to understand and further analyze.

Simplifying Expressions

Simplifying expressions is the final step in polynomial multiplication, where the arithmetic of combining like terms brings us to a cleaner and more manageable expression. Simplifying helps in summarizing our results into the most straightforward form, which is crucial for problem-solving and communication. After combining terms in our exercise, we arrive at the expression \(6x^3 - 5x^2 + 13x - 6\).

• It's important to ensure there are no further like terms to combine.
• Make sure all calculations are double-checked for accuracy.

The simplified form of a polynomial is not only aesthetically pleasing but also efficient for any further computations or evaluations, providing a clear view of each term and its coefficient.