Problem 57 - Algebra A Combined Function

Question

Multiply vertically. \(\left(x^{2}+5 x-7\right)\left(2 x^{2}-7 x-9\right)\)

Step-by-Step Solution

Answer

The product is \(2x^4 + 3x^3 - 58x^2 + 4x + 63.\)

Step 1: Apply the Distributive Property

To multiply the two polynomials \((x^2 + 5x - 7)\) and \((2x^2 - 7x - 9)\), use the distributive property. This means each term in the first polynomial needs to be multiplied by each term in the second polynomial.

Step 2: Multiply \(x^2\) with Each Term of the Second Polynomial

Calculate: \[x^2 \cdot 2x^2 = 2x^4,\] \[x^2 \cdot (-7x) = -7x^3,\] \[x^2 \cdot (-9) = -9x^2.\]

Step 3: Multiply \(5x\) with Each Term of the Second Polynomial

Calculate: \[5x \cdot 2x^2 = 10x^3,\] \[5x \cdot (-7x) = -35x^2,\] \[5x \cdot (-9) = -45x.\]

Step 4: Multiply \(-7\) with Each Term of the Second Polynomial

Calculate: \[-7 \cdot 2x^2 = -14x^2,\] \[-7 \cdot (-7x) = 49x,\] \[-7 \cdot (-9) = 63.\]

Step 5: Combine All the Results

Compile all the terms obtained from Steps 2-4: \[2x^4 - 7x^3 - 9x^2 + 10x^3 - 35x^2 - 45x - 14x^2 + 49x + 63.\]

Step 6: Group Like Terms

Combine like terms to simplify the polynomial: \[2x^4 + (-7x^3 + 10x^3) + (-9x^2 - 35x^2 - 14x^2) + (-45x + 49x) + 63.\]

Step 7: Simplify the Polynomial

Add the coefficients of like terms to get the final result: \[2x^4 + 3x^3 - 58x^2 + 4x + 63.\]

Key Concepts

Distributive Property

The distributive property is a foundational principle in algebra that allows us to multiply two algebraic expressions together. It's like sharing something evenly among several people. When used in polynomial multiplication, each term in the first polynomial is multiplied by each term in the second polynomial. This is often called 'distributing' the terms.

Let's break it down into simple steps:

• Take each term of the first polynomial.
• Multiply it by every term in the second polynomial.
• Make sure to keep track of the sign (positive or negative) for each term.

By doing this, you ensure that every possible combination of terms is accounted for. In our example, \((x^2 + 5x - 7)\) and \((2x^2 - 7x - 9)\), each term of the first polynomial is multiplied by each term of the second polynomial.

Like Terms

Once you have used the distributive property, you'll end up with a lot of terms. Some of these terms might look similar, and that's where 'like terms' come into play. Like terms are terms whose variable parts are the same. They only differ in their coefficients.

For example, \(-7x^3\) and \(10x^3\) are like terms because they both have \(x^3\) as their variable part. Similarly, \(-9x^2\), \(-35x^2\), and \(-14x^2\) are like terms because they share the \(x^2\).

After distributing, group all the like terms together. This makes it easier to combine and simplify the expression later. Identifying like terms is a crucial step because it leads to a simpler polynomial expression.

Simplifying Polynomials

Simplifying polynomials is the final step in the process of polynomial multiplication. It involves combining all the like terms you've identified. This is done by adding or subtracting their coefficients. This step helps in consolidating the expression into a neat, organized form.

To simplify, follow these steps:

• Group all like terms together – for example, all \(x^3\) terms and all \(x^2\) terms.
• Add or subtract the coefficients of each group of like terms.
• Arrange the polynomial in descending powers of the variable for a standardized format.

In our solution, the combination of like terms results in \(2x^4 + 3x^3 - 58x^2 + 4x + 63\). This not only makes the polynomial easier to read but also means it is in its simplest form, ready to use for further calculations or evaluations.