Problem 51 - Algebra 1: Concepts and Skills

Question

$$ (2 x+3)\left(3 x^{2}-4 x+2\right) $$

Step-by-Step Solution

Answer

The product of the expressions is \(6x^3 + x^2 - 8x + 6\)

Step 1: Apply the distributive (FOIL) property

Multiply each term in the first polynomial by every term in the second polynomial. \(FOIL\) stands for First, Outer, Inner, Last. Using this method:- First: Multiply the first term in each binomial: \(2x * 3x^2 = 6x^3\)- Outer: Multiply the outer terms in each binomial: \(2x * 2 = 4x\)- Inner: Multiply the inner terms in each binomial: \(3 * 3x^2 = 9x^2\)- Last: Multiply the last term in each binomial: \(3 * 2 = 6\)

Step 2: Combine like terms

Now combine the individual products obtained in step 1 to get the final answer. Here there are no like terms. So, the final answer will be the straightforward addition of the terms. \(6x^3 - 8x^2 + 4x + 9x^2 - 12x + 6\)

Step 3: Simplify

Rearrange and merge like terms. The final polynomial is \(6x^3 + x^2 - 8x + 6\)

Key Concepts

Distributive Property

To understand polynomial multiplication, begin with the distributive property. This essential algebraic property is at the core of multiplying terms within an expression. It allows you to multiply a single term by two or more terms inside a parenthesis, ensuring each term inside the parenthesis is multiplied by the term outside. The general formula can be expressed as:
\( a(b + c) = ab + ac \).
Applying it to the exercise, you multiply each term in the first polynomial \(2x + 3\) by every term in the second polynomial \(3x^2 - 4x + 2\), distributing the multiplication across the terms.

FOIL Method

A specific case of the distributive property used for binomials is the FOIL method. This mnemonic stands for First, Outer, Inner, and Last, representing the order in which you multiply the terms.

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outer terms in the pair of binomials.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms in each binomial.

After applying the FOIL method, the terms are:
\(2x \times 3x^2 = 6x^3\text{ (First)}\), \(3 \times 3x^2 = 9x^2\text{ (Inner)}\), \(2x \times -4x = -8x^2\text{ (Outer)}\), and \(3 \times 2 = 6\text{ (Last)}\). This orderly approach ensures all terms are accounted for in the multiplication.

Combining Like Terms

Once you've multiplied the polynomials, the next step is to combine like terms. Like terms have the same variable raised to the same power. By combining them, you simplify the expression, making it easier to work with. For instance, \(9x^2\) and \( -8x^2\) from our multiplication results are like terms, summing up to \(x^2\). Combining like terms is a simplification process that follows this principle:
\(ax^n + bx^n = (a + b)x^n\).
So, for the current exercise, \(6x^3 - 8x^2 + 4x + 9x^2 - 12x + 6\) simplifies to \(6x^3 + x^2 - 8x + 6\) by combining like terms.

Algebraic Expressions Simplification

The final step is simplifying the algebraic expression, which includes arranging terms in descending order of their degree and ensuring that like terms are combined.
Always start with the highest power first and progress to the constant. In the exercise, after combining like terms, you already have the expression in descending order: \(6x^3 + x^2 - 8x + 6\).
Simplification doesn't change the value of the expression, it only makes it cleaner and more organized. This final form is useful for further operations, such as calculus, and for understanding the expression's behavior.