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Chapter 1: Problem 22

Find the sum, difference, or product. $$4\left(x^{2}-3 x+5\right)-3\left(x^{2}-2 x+1\right)$$

Short Answer

Expert verified
The simplified expression is \(7x^2 - 18x + 23\).

Step by step solution

01

Distribute the Constants

Begin by distributing the constants in front of each parenthesis across the terms inside the parentheses. This involves using the distributive property for both expressions.For the first expression: \[4(x^2 - 3x + 5) = 4 \cdot x^2 - 4 \cdot 3x + 4 \cdot 5 = 4x^2 - 12x + 20\]For the second expression: \[-3(x^2 - 2x + 1) = -3 \cdot x^2 + 3 \cdot 2x - 3 \cdot 1 = -3x^2 + 6x - 3\]
02

Combine the Distribution Results

Now combine the results from distributing the constants by subtracting the second expression from the first one, according to the given equation:\[(4x^2 - 12x + 20) - (-3x^2 + 6x - 3)\]This means you add the opposite of the second expression.
03

Simplify the Expression

Notice that when distributing the subtraction, the signs of the second polynomial change:\[4x^2 - 12x + 20 + 3x^2 - 6x + 3\]Now, combine like terms:- Combine the \(x^2\) terms: \(4x^2 + 3x^2 = 7x^2\).- Combine the \(x\) terms: \(-12x - 6x = -18x\).- Combine the constant terms: \(20 + 3 = 23\).The simplified expression is:\[7x^2 - 18x + 23\]

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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 algebraic rule that allows you to multiply a single term by several terms inside a set of parentheses. It ensures that every term inside the parentheses gets multiplied by the term outside. In the given exercise, you can see the distributive property at work when we tackled each expression separately:
  • For the first expression, we distributed the constant 4 across \( x^2 - 3x + 5 \), resulting in \( 4x^2 - 12x + 20 \).
  • For the second expression, we distributed the constant -3 across \( x^2 - 2x + 1 \), giving \( -3x^2 + 6x - 3 \).
The distributive property is crucial because it allows you to simplify expressions and solve equations more easily. Always ensure that each term inside the parenthesis is multiplied by the constant outside and keep track of positive and negative signs during the process.
Combining Like Terms
Combining like terms is the process of adding or subtracting terms that have the same variable and exponent. It is an important step when simplifying polynomial expressions to ensure clarity and accuracy.After using the distributive property on both expressions in the problem, you end up with:
  • \(4x^2 - 12x + 20\) from the first distribution
  • \(-3x^2 + 6x - 3\) from the second distribution
We combine like terms by grouping:
  • \(x^2\) terms: \(4x^2\) and \(-3x^2\) are combined to make \(7x^2\).
  • \(x\) terms: \(-12x\) and \(6x\) combine to make \(-18x\).
  • Constant terms: \(20\) and \(-3\) sum up to \(23\).
Combining like terms helps to transform long expressions into a more manageable and understandable form.
Simplifying Expressions
Simplifying expressions involves reducing them to their most concise form without changing their value. It's like cleaning up a space for better understanding and ease of use.In the exercise, after distributing and combining like terms, we get the expression \(7x^2 - 18x + 23\). This is the simplified form. Here鈥檚 why:
  • We applied the distributive property to get rid of parentheses and distribute the constants across the terms.
  • We combined like terms, making the expression tidy and grouped by the power of \(x\).
The simplified expression provides a clear interpretation of the polynomial's composition. Simplifying expressions is useful both for solving problems and for presenting solutions clearly.

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Most popular questions from this chapter

Complete the squares in the general equation \(x^{2}+a x+y^{2}+b y+c=0\) and simplify the result as much as possible. Under what conditions on the coefficients \(a, b,\) and \(c\) does this equation represent a circle? A single point? The empty set? In the case in which the equation does represent a circle, find its center and radius.

Radicals Simplify the expression, and eliminate any negative exponents(s). Assume that all letters denote positive numbers. (a) \(\sqrt{x^{3}}\) (b) \(\sqrt[5]{x^{6}}\)

More on Solving Equations Find all real solutions of the equation. $$\sqrt{11-x^{2}}-\frac{2}{\sqrt{11-x^{2}}}=1$$

Prove the following Laws of Exponents. (a) Law \(6:\left(\frac{a}{b}\right)^{-n}=\frac{b^{n}}{a^{n}}\) (b) Law \(7: \frac{a^{-n}}{b^{-m}}=\frac{b^{m}}{a^{n}}\)

Show that the equation represents a circle, and find the center and radius of the circle. $$3 x^{2}+3 y^{2}+6 x-y=0$$
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