Problem 42 - Algebra A Combined Function

Question

Multiply. \((3-2 a)(2-a)\)

Step-by-Step Solution

Answer

The product is \(2a^2 - 7a + 6\).

Step 1: Distribute First Term

First, we will distribute the first term of the first binomial (3) to each term in the second binomial (2 and \(-a\)). Start by multiplying 3 with 2, which gives us \(3 \times 2 = 6\). Then multiply 3 with \(-a\), giving \(3 \times (-a) = -3a\). Thus, from the first term of the first binomial, we have \(6 - 3a\).

Step 2: Distribute Second Term

Next, distribute the second term of the first binomial (-2a) to each term in the second binomial. Start by multiplying \(-2a\) with 2, which gives us \(-2a \times 2 = -4a\). Then multiply \(-2a\) with \(-a\), giving \(-2a \times (-a) = 2a^2\). Thus, from the second term of the first binomial, we get \(-4a + 2a^2\).

Step 3: Combine Like Terms

Now, combine all the terms from Step 1 and Step 2: \(6 - 3a - 4a + 2a^2\). Combine the like terms \(-3a\) and \(-4a\) to get \(-7a\). Re-organize the expression: \(2a^2 - 7a + 6\).

Step 4: Write the Final Answer

The expression has been simplified to \(2a^2 - 7a + 6\). This is the final result after multiplying the binomials \((3-2a)(2-a)\).

Key Concepts

Distributive Property

In algebra, the distributive property is a fundamental principle used to simplify expressions involving multiplication over addition or subtraction. It states that for any three terms, say \(a\), \(b\), and \(c\), the equation \(a(b + c) = ab + ac\) holds true. This property breaks down complex problems into simpler parts, making multiplication easier to handle.

Let's see how the distributive property is applied in our example. When multiplying the binomials \((3 - 2a)(2 - a)\), we individually distribute the terms from the first binomial \((3 - 2a)\) to the second binomial \((2 - a)\).

• First, we multiply 3 by each term of the second binomial: \(3 \times 2 = 6\) and \(3 \times (-a) = -3a\).
• Next, distribute \(-2a\): \(-2a \times 2 = -4a\) and \(-2a \times (-a) = 2a^2\).

Applying the distributive property helps us separate and rearrange these terms, making it easier to solve the entire expression.

Polynomial Expansion

Polynomial expansion involves multiplying two or more polynomials together to simplify or extend an expression. In our problem, we have two binomials: \((3 - 2a)\) and \((2 - a)\). When multiplied, these binomials expand into a new polynomial using the distributive property we discussed earlier.

The purpose of polynomial expansion is to convert a factored form into an extended additive or subtractive form. This helps in analyzing and understanding the polynomial with all its individual terms.

In our example:

• We started with the expression \((3 - 2a)(2 - a)\).
• By expanding, it turns into \(6 - 3a - 4a + 2a^2\).

Each multiplication step unraveled the terms that were concealed in the brackets, showing how these numbers operate together.

Like Terms

Like terms are terms in an algebraic expression that have identical variable parts. This means the terms must have the same variable raised to the same power. Recognizing and combining like terms is a vital step in simplifying algebraic expressions after expansion.

For instance, after expanding our binomials \((3 - 2a)(2 - a)\), we obtained the expression \(6 - 3a - 4a + 2a^2\). Here, \(-3a\) and \(-4a\) are like terms because they both contain the variable \(a\) raised to the same power.

• Combine \(-3a\) and \(-4a\) to get \(-7a\).

This step reduces the expression to \(2a^2 - 7a + 6\).

Combining like terms tidies up our expression, leading to a simpler, clearer form, and ultimately allows us to reach the final polynomial result efficiently.