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Chapter 4: Problem 118

Simplify. $$-5\left(-2 x^{2}-3 x+7\right)$$

Short Answer

Expert verified
The simplified form of the given expression is \(10 x^{2} +15x - 35\).

Step by step solution

01

Apply Distributive Property

Multiply -5 with each term inside the bracket. The calculation will look like this: \(-5 * (-2 x^{2}) + (-5 * -3 x) + (-5 * 7)\). Remember that a negative times a negative gives a positive result and a negative times a positive gives a negative result.
02

Multiply The Coefficients

The next step is to actually perform the multiplication. This results in the following: \(10 x^{2} +15x - 35\)
03

Write The Final Simplified Expression

The final expression is a simplified polynomial. Write it as: \(10 x^{2} +15x - 35\). There are no further possible simplifications, we can stop here.

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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 fundamental concept in mathematics, especially when simplifying expressions, including polynomials. It allows you to multiply a single term by each term within a parenthesis or bracket. Here, the expression is \(-5(-2x^2 - 3x + 7)\).
  • The distributive property tells us to take the \(-5\) outside the brackets and multiply it by each term inside: \(-5 imes (-2x^2)\), \(-5 imes -3x\), and \(-5 imes 7\).
  • Notice how we systematically go through each term one by one, ensuring each is multiplied correctly.
  • Remember, when multiplying, the rule of signs holds: A negative times a negative yields a positive, and a negative times a positive yields a negative result.
Using this property ensures that all terms are accounted for and helps us move towards simplifying the expression entirely.
Polynomial Expression
Understanding what a polynomial expression is can help greatly in simplifying it. Polynomials are expressions composed of variables and coefficients, which are operated on by addition, subtraction, and multiplication, but never division by a variable. In this exercise, we're working with a polynomial inside brackets: \(-2x^2 - 3x + 7\).
  • The polynomial consists of three terms: a quadratic term \(-2x^2\), a linear term \(-3x\), and a constant \(7\).
  • Simplifying polynomials involves combining like terms, but in this case, we're more focused on correctly distributing the factor (in this case, \(-5\)) to each term.
  • Each term in a polynomial is distinct, identified by the degree of the variable it contains. The degree is the highest power of the variable in that term.
Knowing the structure of a polynomial makes it clearer to see when an expression is fully simplified or when further action is needed.
Multiplication of Coefficients
Once the distributive property has been applied, and we've correctly set up our expression for multiplication, the next step is multiplying coefficients. This means taking the numerical coefficient (the number in front of the variable), and multiplying it by the given factor. In this exercise, that factor is \(-5\).
  • For \(-5 imes -2x^2\), multiply the coefficients \(-5\) and \(-2\) to get \(10\), resulting in \(10x^2\).
  • For \(-5 imes -3x\), \(-5\) multiplied by \(-3\) gives \(15\), resulting in \(15x\).
  • Finally, for \(-5 imes 7\), \(-5\) times \(7\) results in \(-35\).
Remember, keeping a close eye on the signs as you multiply will ensure accurate results. The final output, \(10x^2 + 15x - 35\), confirms that all coefficients were multiplied correctly, and the expression is now simplified.

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