Problem 56 - Algebra A Combined Function

Question

Multiply vertically. \((4 x-5)\left(8 x^{2}+2 x-4\right)\)

Step-by-Step Solution

Answer

The product is \(32x^3 - 32x^2 - 26x + 20\).

Step 1: Distribute the First Term of Binomial

Start by multiplying the first term of the binomial \((4x - 5)\), which is \(4x\), by each term in the trinomial \((8x^2 + 2x - 4)\). Calculate:- \(4x \cdot 8x^2 = 32x^3\)- \(4x \cdot 2x = 8x^2\)- \(4x \cdot (-4) = -16x\)This gives the result: \[32x^3 + 8x^2 - 16x\]

Step 2: Distribute the Second Term of Binomial

Now multiply the second term of the binomial \((4x - 5)\), which is \(-5\), by each term in the trinomial \((8x^2 + 2x - 4)\).Calculate:- \(-5 \cdot 8x^2 = -40x^2\)- \(-5 \cdot 2x = -10x\)- \(-5 \cdot (-4) = 20\)This gives the result: \[-40x^2 - 10x + 20\]

Step 3: Combine Like Terms

Now combine the results from Step 1 and Step 2. Combine like terms:- \(32x^3\) remains unchanged- \(8x^2 - 40x^2 = -32x^2\)- \(-16x - 10x = -26x\)- The constant \(20\) remains unchangedThus, the expression simplifies to:\[32x^3 - 32x^2 - 26x + 20\]

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle used in polynomial multiplication. It involves distributing, or sharing, one term across the terms of another expression. This property states that for any numbers or variables

• a,
• b,
• c,

The following holds true:

• a(b + c) = ab + ac.

To apply this property in the polynomial multiplication of

• (4x - 5) and (8x^2 + 2x - 4),

the first step is to distribute each term in the binomial (4x - 5) across the trinomial (8x^2 + 2x - 4). The distributive property ensures that you account for each combination of terms from the two polynomials. By multiplying every term in one polynomial by every term in the other, you eventually capture every interaction between these expressions. This approach prepares you for the next crucial steps in combining and simplifying the expressions into a final, manageable polynomial.

Like Terms

In algebra, like terms are terms that have identical variable parts raised to the same power. For example, in the expression

• 8x^2 - 40x^2,

these two are like terms because they both have the variable x raised to the second power. The importance of like terms lies in their ability to be combined for simplification. To achieve this, you add or subtract the coefficients while keeping the variable part unchanged. In the exercise, combining like terms after distribution involves:

• keeping terms like 32x3 as they are because no other
  • 'x³'
  terms exist,
• adding
  • 8x² - 40x² = -32x²
  by subtracting the coefficients,
• combining
  • -16x - 10x = -26x
  by subtracting the coefficients for x terms, and
• bringing down constant terms like 20 without any combinations.

Recognizing and working with like terms is essential for streamlining expressions and solving polynomial equations efficiently.

Binomial

A binomial is a polynomial with exactly two terms. These terms can be anything, as long as there are two. Common forms include:

• a + b
• x - y
• 4x - 5,

like the one in the exercise. Binomials are a stepping stone into more complex algebra because they introduce the concept of combining like terms when expanded. In polynomial multiplication, understanding a binomial’s role is critical because you must ensure each part of the binomial is distributed correctly within the context of a problem.

During multiplication, you'll find it important to treat each term in the binomial individually. In the problem (4x - 5)(8x² + 2x - 4), both 4x and -5 need to separately interact with each term in the trinomial. This captures how each portion of the binomial contributes to forming a larger polynomial. Practicing with binomials helps develop the skills needed to handle more extensive expressions.

Trinomial

Trinomials are polynomials that consist of three terms. They might look like:

• x² + y + z
• 8x² + 2x - 4,

as featured in the exercise. These are more complex than binomials, offering additional opportunities for simplifying and rearranging algebraic expressions.

When multiplying a trinomial like (8x² + 2x - 4) by a binomial, it’s integral to apply the distributive property to ensure each pair of terms interact. Each term of the binomial must multiply every term in the trinomial, ensuring nothing is left unaccounted.

• This results in more products to manage and combine, emphasizing the need for a systematic approach in algebra.
• Each pair product (e.g., the result of 4x and 8x² or -5 and -4) should be carefully calculated.
• Keep track of signs and ensure the correct combination of like terms.

Understanding trinomials’ behavior in expressions allows you to not only simplify the exercise but also apply these skills to solve equations involving roots and factors of expressions.