Question

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

Short answer

The result of the subtraction is \( 4x^2 - 4x - 5 \).

Step-by-step solution

Align Polynomials Vertically

Firstly, align the given two polynomials vertically on the basis of their powers. Each term with the same power of x in the first polynomial is written above its corresponding term in the second polynomial: \n\[\begin{align*}& 6x^2 -3x + 2 \& - (2x^2 + x + 7) \\end{align*}\]

Apply Subtraction

Next, apply subtraction term by term, that is subtract each term in the second polynomial from the corresponding term in the first polynomial: \n\[\begin{align*}& (6x^2 - 2x^2) + (-3x - x) + (2 - 7)\& = 4x^2 - 4x - 5\end{align*}\] Where, 6x^2 minus 2x^2 gives 4x^2, -3x minus x gives -4x, 2 minus 7 gives -5.

State the Result

The final result of subtracting the second polynomial from the first is 4x^2 - 4x - 5, which is the answer.

Key concepts

Polynomial Subtraction

Polynomial subtraction might at first sound complex, but it's essentially a straightforward process of subtracting each term of one polynomial from the corresponding term of another polynomial. Think of it like simplifying a complex sentence by breaking it down into easier words. When we talk about polynomials, we're looking at algebraic expressions that can have multiple terms, each consisting of variables raised to a power (usually whole numbers) and multiplied by coefficients.

For example, when we have two polynomials such as \(6x^2 - 3x + 2\) and \(2x^2 + x + 7\), subtracting them involves taking away the terms of the second polynomial from the first one. An important rule to remember is that we can only subtract terms with the same degree, meaning terms with the same power of the variable.

Aligning Like Terms

Before you can add or subtract polynomials effectively, you have to align like terms. What are like terms, then? They are terms that have the same variable raised to the same power. In the case of our example polynomials \(6x^2 - 3x + 2\) and \(2x^2 + x + 7\), the like terms are the terms that both have \(x^2\), the ones that have \(x\) to the power of 1, and the constant terms.

To ensure we subtract correctly, we write these terms one below the other, just as we would in a column when doing a long subtraction with numbers. This makes it easy to match and subtract the corresponding coefficients properly, ensuring the subtraction is accurately carried out term by term.

Vertical Subtraction Method

The vertical subtraction method is a systematic way to subtract polynomials, ensuring you line up the corresponding terms. It's similar to the method you would use for traditional long-hand numerical subtraction. After you've aligned like terms in a column, you apply the subtraction operation for each pair of like terms.

Here is a breakdown of the steps using our example polynomials:\(6x^2 - 3x + 2\) and \(2x^2 + x + 7\):

• First, you write one polynomial above the other ensuring like terms are vertically aligned.
• Second, subtract the lower polynomial from the upper polynomial term by term.
• Finally, write down the result beneath each column for your final answer.

Applying these steps would result in \(4x^2 - 4x - 5\), a simplified expression that represents the difference between the two polynomials.