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To solve polynomial multiplication, we use the distributive property. This property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products.
In our example, we need to multiply each term in one polynomial by each term in the other polynomial.
Let's break this down further. For the polynomial \((2x - 1)(3x^5 - 2x^3 + x^2 - 2x + 3)\), you start by distributing the first term, which is \(2x\).
That means \(2x\) will be multiplied by every term inside the parentheses:

• \(2x \times 3x^5 = 6x^6\)
• \(2x \times (-2x^3) = -4x^4\)
• \(2x \times x^2 = 2x^3\)
• \(2x \times (-2x) = -4x^2\)
• \(2x \times 3 = 6x\)

Similarly, distribute the second term \(-1\) by multiplying it with each term inside the parentheses:

• \(-1 \times 3x^5 = -3x^5\)
• \(-1 \times (-2x^3) = 2x^3\)
• \(-1 \times x^2 = -x^2\)
• \(-1 \times (-2x) = 2x\)
• \(-1 \times 3 = -3\)

This step sets the foundation for the rest of the solution. Understanding this property is key to successfully multiplying polynomials.

Once you've distributed both terms, the next step is combining like terms. Like terms are terms that contain the same variables raised to the same power.
In our example from the previous step, we’re left with:

• \(6x^6\)
• \(-3x^5\)
• \(-4x^4\)
• \(2x^3 + 2x^3 = 4x^3\)
• \(-4x^2 - x^2 = -5x^2\)
• \(6x + 2x = 8x\)
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Lastly, it’s important to understand algebraic expressions when dealing with polynomial multiplication. Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (e.g., addition, subtraction, multiplication, division).
In our original problem \((2x - 1)(3x^5 - 2x^3 + x^2 - 2x + 3)\), we are working with a product of two algebraic expressions.
When multiplying these expressions, each term in the first polynomial interacts with each term in the second polynomial.
It is essential to keep track of each multiplication and then combine the like terms properly. By understanding and mastering algebraic expressions, you can simplify and solve more complex polynomial multiplications effectively. This will be especially helpful for higher-level algebra courses and standardized tests.