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Chapter 5: Problem 24

Subtract. $$ \left(-7 x^{2}+4 x+7\right)-(-8 x+2) $$

Short Answer

Expert verified
\(-7x^2 + 12x + 5\)

Step by step solution

01

Distribute the Negative Sign

First, distribute the negative sign across the second polynomial. This changes each sign in the second polynomial. So, \(-(-8x+2)\) becomes \(+8x-2\).
02

Rewrite the Expression

Replace the original subtraction with the equivalent addition calculated in Step 1. The expression now becomes:\(-7x^2 + 4x + 7 + 8x - 2\).
03

Combine Like Terms

Identify and combine like terms in the polynomial. The like terms are:- \(4x + 8x = 12x\) - \(7 - 2 = 5\)So the expression simplifies to \(-7x^2 + 12x + 5\).
04

Final Expression

After simplifying, the polynomial subtraction results in the final expression:\(-7x^2 + 12x + 5\).

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Key Concepts

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

Distributive Property
The distributive property is a key mathematical rule that's extremely useful when working with expressions, especially in tasks like polynomial subtraction. In simple terms, it involves distributing a single factor across terms inside a parenthesis. When you have something like a number or a negative sign outside, you distribute it to each term inside.

Here's the basic idea: \( a(b + c) = ab + ac \). Let's see it in action in polynomial subtraction where we focus on the negative sign:

  • You saw \( -(8x + 2) \).
  • Apply the distributive property to change each term's sign: \( +8x - 2 \).

Using this property correctly helps simplify complex expressions and is fundamental in rearranging terms properly for subtraction.
Like Terms
Understanding like terms is crucial when working with polynomials. Like terms are terms that have the exact same variable raised to the same power. Only these can be combined. This makes expressions more manageable and simpler. Let's break it down further:

  • Consider the expression \( -7x^2 + 4x + 8x \).
  • Recognize that \( 4x \) and \( 8x \) are like terms (both have \( x \) as their variable).
  • Combine them: \( 4x + 8x = 12x \).

  • Constants are also like terms, such as \( 7 \) and \( -2 \), which combine to \( 5 \).

By understanding and identifying like terms, solving polynomial problems becomes a much simpler process, ensuring accurate results.
Polynomial Expressions
Polynomial expressions are expressions comprising variables such as \( x \), coefficients (numbers multiplied by these variables), and constants. These elements work together to form terms. A polynomial's degree is determined by its highest power of a variable.

Let's delve into a simple polynomial expression from the example given: \( -7x^2 + 12x + 5 \). Here's how it's structured:

  • \

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Most popular questions from this chapter

Add or subtract as indicated. $$ \left(-a^{2}+1\right)-\left(a^{2}-3\right)+\left(5 a^{2}-6 a+7\right) $$

Simplify each expression by performing the indicated operation. Explain how you arrived at each answer. a. \(z+3 z\) b. \(z \cdot 3 z\) c. \(-z-3 z\) d. \((-z)(-3 z)\)

Add \(\left(4 x^{2}-6 x+1\right)\) and \(\left(3 x^{2}+2 x+1\right)\)

Simplify each expression by performing the indicated operation. Explain how you arrived at each answer. a. \(2 y+y\) b. \(2 y \cdot y\) c. \(-2 y-y\) d. \((-2 y)(-y)\)

Add or subtract as indicated. $$ \left(3 x^{2} y-6 x y+x^{2} y^{2}-5\right)-\left(11 x^{2} y^{2}-1+5 y x^{2}\right) $$
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