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Chapter 5: Problem 104

Subtract. $$ \left(4 y^{2}-13 y z-9 z^{2}\right)-\left(9 y^{2}-6 y z+3 z^{2}\right) $$

Short Answer

Expert verified
-5y^2 - 7yz - 12z^2

Step by step solution

01

Write down the given expression

Start with the given expression: i o o (4y^2 - 13yz - 9z^2) - (9y^2 - 6yz + 3z^2)Next, we will distribute the negative sign across the second expression.
02

Distribute the negative sign

Distribute the negative sign to each term in the second expression: o o o (4y^2 - 13yz - 9z^2) - 9y^2 + 6yz - 3z^2Rewrite the expression after distributing the negative sign.
03

Combine like terms

Combine the like terms in the expression: o o o (4y^2 - 9y^2) + (-13yz + 6yz) + (-9z^2 - 3z^2)Calculate the resulting coefficients for each term.
04

Simplify each term

Simplify each term: o o o (4y^2 - 9y^2) = -5y^2 o (-13yz + 6yz) = -7yz o (-9z^2 - 3z^2) = -12z^2
05

Write the final answer

Combine all simplified terms to write the final answer: o -5y^2 - 7yz - 12z^2

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributing Negative Sign
When subtracting polynomials, one important step is distributing the negative sign (-) to each term in the polynomial being subtracted. This means we change the signs of all terms in the second polynomial before carrying out the subtraction.

Let's look at the exercise: \( (4y^{2} - 13yz - 9z^{2}) - (9y^{2} - 6yz + 3z^{2}) \).
We need to distribute the negative sign across \( (9y^{2} - 6yz + 3z^{2}) \).

This changes each term's sign:
  • Positive 9y虏 becomes negative, making it -9y虏.
  • Negative 6yz becomes positive, making it +6yz.
  • Positive 3z虏 becomes negative, making it -3z虏.

Our expression looks like this now:
\( 4y^{2} - 13yz - 9z^{2} - 9y^{2} + 6yz - 3z^{2} \).
By distributing the negative sign, we prepare the polynomials for combining like terms.
Combining Like Terms
After distributing the negative sign, the next step is to combine like terms.
Like terms are terms that contain the same variables raised to the same power.

In the expression \( 4y^{2} - 13yz - 9z^{2} - 9y^{2} + 6yz - 3z^{2} \), we look for:
  • Terms with \( y^2 \): \( 4y^{2} \) and \( -9y^{2} \)
  • Terms with \( yz \): \( -13yz \) and \( +6yz \)
  • Terms with \( z^2 \): \( -9z^{2} \) and \( -3z^{2} \)

We group them together and perform the operations:
  • \

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Most popular questions from this chapter

A rectangular piece of tin is twice as long as it is wide. Squares \(2 \mathrm{cm}\) on a side are cut out of each corner, and the ends are turned up to make a box whose volume is \(480 \mathrm{cm}^{3} .\) What are the dimensions of the original piece of tin? (GRAPH CANNOT COPY)

Factor completely. $$ -16+17\left(5-y^{2}\right)-\left(5-y^{2}\right)^{2} $$

Perform the indicated operations. $$ \left(8 r^{2}-6 r\right)-(2 r-6)+\left(5 r^{2}-7\right) $$

Solve. Postage Rates. The maximum size envelope that can be mailed at the Large Envelope rate is 3 in. longer than it is wide. The area is 180 in \(^{2} .\) Find the length and the width.

Solve. Three consecutive even integers are such that the square of the first plus the square of the third is \(136 .\) Find the three integers.
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