Problem 37 - Algebra 1

Question

SUBTRACTING VERTICALLY Use a vertical format to subtract the second polynomial from the first polynomial. $$4 x^{3}+3 x^{2}+8 x+6,2 x^{3}-3 x^{2}-7 x$$

Step-by-Step Solution

Answer

After subtracting the second polynomial from the first polynomial, the result is \(2 x^{3} + 6 x^{2}+ x + 6\).

Step 1: Aligning Polynomials

Write the first and second polynomials vertically and align terms according to the degree of each term. \[\begin{array}{c}4 x^{3}+3 x^{2}+8 x+6 \- (2 x^{3}-3 x^{2}-7 x)\\end{array}\]

Step 2: Distribute Negative Sign

First distribute the negative sign to each term in the second polynomial: \[\begin{array}{c}4 x^{3}+3 x^{2}+8 x+6 \-2 x^{3}+3 x^{2}+7 x\\end{array}\]

Step 3: Subtract Term by Term

Now, subtract each term in the second polynomial from the corresponding term in the first polynomial. This gives: \( (4 x^{3} - 2 x^{3})+ (3 x^{2} + 3 x^{2})+ (8 x - 7 x) + 6 = 2 x^{3} + 6 x^{2}+ x + 6 \)

Key Concepts

Vertical Format

When subtracting polynomials, organizing them in a vertical format is an essential first step. This approach helps ensure that all like terms are correctly aligned. After listing the polynomials vertically, arrange them so that like terms with the same degree are directly above or below each other. This format is akin to arithmetic subtraction, such as subtracting long numbers, where it's crucial to line up each digit. The highest degree term should be placed in front, descending to the lowest degree. Once aligned correctly, the subtraction process can proceed without confusion, allowing for accurate subtraction right down the columns.

Aligning Polynomials

Aligning polynomials is a critical step when performing a vertical subtraction, primarily to avoid errors and simplify the problem-solving process. Place each polynomial in columns based on their degrees:

• The terms like \(4x^3\) and \(2x^3\) align because both are cubic terms.
• The quadratic terms, like \(3x^2\) and \(-3x^2\), must be directly lined up together.
• Similarly, align linear terms like \(8x\) and \(-7x\).
• Constant terms, such as \(6\), should align with other constants or remain unpaired if they are only present in one polynomial.

This accurate alignment is crucial for distributing the negative signs and for simplifying the subtraction process.

Distributing Negative Signs

Distributing the negative signs is a pivotal task in polynomial subtraction. When subtracting one polynomial from another, every term in the second polynomial must have its sign changed. This adjustment is because subtraction is akin to adding the opposite. Take, for instance, the expression \(-(2x^3 - 3x^2 - 7x)\):

• \(2x^3\) becomes \(-2x^3\)
• \(-3x^2\) turns into \(+3x^2\)
• \(-7x\) switches to \(+7x\)

Implementing this transformation ensures that when you proceed with term-by-term subtraction, the calculations are accurate. This step essentially turns subtraction into a form of addition, simplifying the operation.

Term by Term Subtraction

After distributing negative signs, it is time to perform a term by term subtraction. This means you subtract each term in the newly adjusted second polynomial (with distributed signs) from the corresponding term in the first polynomial:

• For the cubic terms: \(4x^3 - 2x^3 = 2x^3\)
• For the quadratic terms: \(3x^2 + 3x^2 = 6x^2\)
• For the linear terms: \(8x + 7x = 1x\)
• Evaluate any constant term on its own, such as \(6\), from the first polynomial if it stands alone.

Each of these calculations corresponds to a direct column subtraction, similar to computing each vertical slice in a long subtraction problem. The result from this step directly gives the simplified polynomial after subtraction.