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Chapter 6: Problem 51

Find the sum of \(\left(2 p^{3}-8\right)\) and \(\left(p^{2}+9 p+18\right)\).

Short Answer

Expert verified
The sum is \( 2p^3 + p^2 + 9p + 10 \).

Step by step solution

01

- Identify the expressions

Identify the two expressions we need to sum: \(\text{Expression A:}\text{ } 2p^3 - 8\) and \(\text{Expression B:}\text{ } p^2 + 9p + 18\)
02

- Write down the sum of the expressions

Write down the sum of the two expressions: \( (2p^3 - 8) + (p^2 + 9p + 18) \)
03

- Combine like terms

Combine like terms by adding the coefficients of the same power of \( p \): \( 2p^3 + p^2 + 9p + (-8 + 18) \). Simplify to get:\( 2p^3 + p^2 + 9p + 10 \)

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Key Concepts

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

Combining Like Terms
When adding polynomials, one crucial step is combining like terms. Like terms are terms that have the same variables and exponents. For example, in the polynomial expressions we are adding, terms like \(2p^3\) and \(p^2\) are 'like' terms because they have the same variable.
Identifying and combining like terms simplifies the overall expression. Remember to keep the sign in front of the term when combining, such as turning \(9p + (-3p)\) into \(6p\). Combining correctly ensures accuracy in your results.
  • Example: \(4x + 5x = 9x\)
  • Combine \(3p^2 \) and \(4p^2 \) to get \(7p^2\)
  • Combine \(-7 + 3\) to get \(-4\)
Coefficients
Coefficients are the numerical factors in terms of a polynomial. In the expression \(5x + 3y\), 5 and 3 are the coefficients. They are crucial when adding polynomials because they tell you how many of each term you have.
When combining like terms, add or subtract the coefficients to get the new term. For instance, in \(2x + 3x\), the coefficients 2 and 3 add up to give 5x. Always ensure you handle coefficients properly, especially combining terms involving subtraction or negative coefficients.
  • Example: In \(4a - 2a\), subtract the coefficients to get \(2a\)
  • For \(x + 5x\), add the coefficients to get \(6x\)
  • In the term \(-3p\), -3 is the coefficient.
Polynomial Expressions
A polynomial expression contains variables and coefficients, involving terms with non-negative integer exponents. For instance, \(2p^3 - 8\) and \(p^2 + 9p + 18\) are polynomial expressions.
Understanding what a polynomial is crucial to learning how to add them. Each polynomial term must be understood separately by its power and expanded to see the full expression. Simplifying often involves several terms with varying powers.
  • Example: \(x^3 + x^2 + x + 1\) is a polynomial.
  • \(5y^2 - 2y + 3\) is another polynomial.
Simplifying Expressions
Simplifying expressions means reducing them to their simplest form. It often involves combining like terms and adding or subtracting coefficients appropriately. For example, in \(2p^3 + p^2 + 9p + 10 \), each term is already simplified.
Simplification makes the polynomial more understandable and easier to work with. After combining like terms and handling coefficients, double-check your final expression to ensure no further simplification is possible.
  • Simplify \(3x + 4 + 5x\) = \(8x + 4\).
  • Combine \(2y^2 - y + 3y^2\) to get \(5y^2 - y\).

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Most popular questions from this chapter

Mixed Practice (a) \(\frac{z^{6}}{z^{5}}\) (b) \(\frac{z^{5}}{z^{6}}\)

Divide the monomials. $$\frac{\left(4 u^{2} v^{5}\right)\left(15 u^{3} v\right)}{\left(12 u^{3} v\right)\left(u^{4} v\right)}$$

Divide each polynomial by the monomial. $$\left(16 y^{2}-20 y\right) \div 4 y$$

Divide each polynomial by the monomial. $$\frac{4 w^{2}+2 w-5}{2 w}$$

Divide each polynomial by the monomial. $$\frac{10 x^{2}+5 x-4}{-5 x}$$
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