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Chapter 5: Problem 59

Perform each indicated operation. $$ \left(8 m^{2}-7 m\right)-\left(3 m^{2}+7 m-6\right) $$

Short Answer

Expert verified
The result is \( 5m^2 - 14m + 6 \).

Step by step solution

01

- Distribute the Negative Sign

Subtract the entire second polynomial by distributing the negative sign inside the parentheses: \[ (8m^2 - 7m) - (3m^2 + 7m - 6) = 8m^2 - 7m - 3m^2 - 7m + 6 \]
02

- Combine Like Terms

Group the like terms together: \[ (8m^2 - 3m^2) + (-7m - 7m) + 6 \]
03

- Simplify Each Group

Simplify by performing the operations within each group: \[ 8m^2 - 3m^2 = 5m^2 \] \[ -7m - 7m = -14m \] \[ \text{Constant term is 6} \]
04

- Write the Final Expression

Combine the simplified terms to get the final result: \[ 5m^2 - 14m + 6 \]

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Key Concepts

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

Distribute Negative Sign
When we subtract polynomials, it's important to distribute the negative sign throughout the polynomial. This means we need to change the signs of each term within the second polynomial before combining it with the first. In our example, \(3m^2 + 7m - 6\) becomes \(-3m^2 - 7m + 6\) when the negative sign is distributed.
This simplifies the expression, making it easier to combine like terms.
Combine Like Terms
To combine like terms, group together all terms with the same variable and the same exponent. In our exercises, we have:
\(8m^2 - 3m^2\)
\(-7m - 7m\)
The constant term \6\
Like terms have the same variable parts, which allow us to add or subtract their coefficients.
Simplify Polynomials
Simplifying polynomials involves performing arithmetic operations within each group of like terms. For example:
\(8m^2 - 3m^2 = 5m^2\)
\(-7m - 7m = -14m\)
The constant term remains \6\.
Reducing each group to a single term makes it easier to combine the simplified results in the final step.
Algebra Operations
To complete polynomial subtraction, we apply various algebra operations systematically:
  • Distribute: Change the signs of terms in the polynomial being subtracted.
  • Combine: Group and combine like terms.
  • Simplify: Perform arithmetic within each group of like terms.
  • Write: Combine simplified terms to get the final expression.
After following these steps, we get the final expression: \(5m^2 - 14m + 6\).
This method ensures clarity and accuracy.

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Most popular questions from this chapter

Perform each division using the "long division" process. $$ \frac{27 r^{4}-36 r^{3}-6 r^{2}+26 r-24}{3 r-4} $$

Evaluate. $$ 10,000(36.94) $$

Simplify by writing each expression wth positive exponents. Assume that all variables represent nonzero real numbers. $$ \frac{\left(4^{-1} a^{-1} b^{-2}\right)^{-2}\left(5 a^{-3} b^{4}\right)^{-2}}{\left(3 a^{-3} b^{-5}\right)^{2}} $$

The special product $$ (x+y)(x-y)=x^{2}-y^{2} $$ $$ \text { can be used to perform some multiplication problems. Here are two examples.} $$ $$ \begin{aligned} 51 \times 49 &=(50+1)(50-1) \\ &=50^{2}-1^{2} \\ &=2500-1^{2} \\ &=2499 \end{aligned} \quad | \begin{aligned} 102 \times 98 &=(100+2)(100-2) \\ &=100^{2}-2^{2} \\ &=10,000-4 \\ &=9996 \end{aligned} $$ Once these patterns are recognized, multiplications of this type can be done mentally. Use this method to calculate each product mentally. $$ 103 \times 97 $$

Evaluate each expression for \(x=3\) $$ 4 x^{3}-5 x^{2}+2 x-5 $$
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