Problem 58 - Elementary and Intermediate Algebra

Question

Use vertical form to subtract the polynomials. $$ \text { Subtract } 3 y^{2}-6 y+7 \text { from } 6 y^{2}+4 y+13 $$

Step-by-Step Solution

Answer

The result of the subtraction is \(3y^2 + 10y + 6\).

Step 1: Understand the Problem

We need to subtract the polynomial \(3y^2 - 6y + 7\) from the polynomial \(6y^2 + 4y + 13\). This involves aligning and subtracting like terms.

Step 2: Write the Polynomials Vertically

Write the polynomials one below the other, aligning like terms:\[\begin{array}{r}6y^2 + 4y + 13 \-(3y^2 - 6y + 7)\end{array}\]

Step 3: Distribute the Negative Sign

Distribute the negative sign to each term of the polynomial being subtracted:\(- (3y^2 - 6y + 7) = -3y^2 + 6y - 7\)

Step 4: Subtract Term by Term

Now subtract each term in the vertical arrangement:\[\begin{array}{r}6y^2 + 4y + 13 \-3y^2 + 6y - 7\hline3y^2 + 10y + 6\end{array}\]Subtract the coefficients of like terms: - \(6y^2 - 3y^2 = 3y^2\)- \(4y + 6y = 10y\)- \(13 - 7 = 6\)

Step 5: Write the Result

The result of the subtraction is the polynomial \(3y^2 + 10y + 6\).

Key Concepts

Vertical Form

When subtracting polynomials, placing them in a vertical form is incredibly helpful. This method is similar to how you might subtract numbers in elementary arithmetic. You align corresponding terms so that you can directly subtract them from one another.
If we need to subtract the polynomial \(3y^2 - 6y + 7\) from \(6y^2 + 4y + 13\), we start by writing them one above the other.
Make sure that you:

• Line up terms with the same powers of \(y\) in columns.
• Leave space for any missing degrees; for instance, if there is no \(y^1\) term, leave a blank space in that column.

By organizing the polynomials vertically, it becomes much easier to subtract each column of like terms directly.

Like Terms

Like terms in polynomials are terms that have the same variable raised to the same power. Only the coefficients of these terms differ. Identifying like terms is key when performing operations such as addition or subtraction among polynomials.
For example, in our problem, the terms like \(3y^2\) and \(6y^2\) are like terms because they both involve \(y^2\). Similarly, \(-6y\) and \(4y\) involve the same variable to the power of one, making them like terms as well.
When you align polynomials vertically, ensure like terms are in the same column. This way, you subtract the coefficients of these terms directly, just as the numbers are subtracted when lined up by place value.

Distributing Negative

Subtracting polynomials involves a crucial step: distributing the negative sign over the polynomial to be subtracted. This means that you change the signs of each term in the polynomial you are subtracting.
In our given problem, we start by writing the expression \(- (3y^2 - 6y + 7)\). To distribute the negative sign, we change each term's sign:

• \(+3y^2\) becomes \(-3y^2\).
• \(-6y\) becomes \(+6y\).
• \(+7\) becomes \(-7\).

By doing so, you transform the subtraction problem into addition but with all terms in the second polynomial negated. This step simplifies the subtraction process, as you can now subtract each term as a straightforward addition of its negated term.