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Chapter 5: Problem 46

Add. \begin{array}{l} {4 a^{3}-4 a^{2}-4} \\ {6 a^{3}+5 a^{2}-8} \\ \hline \end{array}

Short Answer

Expert verified
The sum is \(10a^{3} + a^{2} - 12\).

Step by step solution

01

- Write Down the Problem

We need to add the two polynomials given: 1. \(4a^{3} - 4a^{2} - 4\) 2. \(6a^{3} + 5a^{2} - 8\)
02

- Align Like Terms

Write the polynomials so that like terms are aligned vertically:\(4a^{3} - 4a^{2} - 4\) \(6a^{3} + 5a^{2} - 8\)
03

- Add the Coefficients of Like Terms

Add the coefficients of the like terms from both polynomials:1. For \(a^{3}\): \(4a^{3} + 6a^{3} = 10a^{3}\)2. For \(a^{2}\): \(-4a^{2} + 5a^{2} = a^{2}\)3. For the constant term: \(-4 - 8 = -12\)
04

- Write the Resulting Polynomial

Combine the results from Step 3 into a single polynomial: \(10a^{3} + a^{2} - 12\)

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Key Concepts

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

Combining Like Terms
When dealing with polynomial addition, combining like terms is a crucial step. Like terms are terms that have the same variable raised to the same power.
For instance, among the terms: \(4a^{3} - 4a^{2} - 4\) and \(6a^{3} + 5a^{2} - 8\), both \(4a^{3}\) and \(6a^{3}\) are like terms because they include the variable \(a\) raised to the power of 3.
Combining like terms involves adding or subtracting their coefficients while keeping the common variable and its exponent unchanged.
  • Example: For V\(a^{3}\): Adding \(4a^{3}\) and \(6a^{3}\) gives \(10a^{3}\).

  • For \(a^{2}\): Adding \(-4a^{2}\) and \(5a^{2}\) gives \(a^{2}\), which is the same as \(1a^{2}\).

  • For constants: Combining \(-4\) and \(-8\) gives \(-12\).
Properly combining like terms simplifies the polynomial, making it easier to understand and work with.
Polynomials
A polynomial is an algebraic expression that consists of variables, coefficients, and exponents.
Polynomials can have multiple terms, and each term is a product of a coefficient and a variable raised to a non-negative integer power.
For example, in the polynomial \(4a^{3} - 4a^{2} - 4\), there are three terms: \(4a^{3}\), \(-4a^{2}\), and \(-4\).
Polynomials can be classified based on the number of terms they have:
  • Monomials: A single term (e.g., \(5x\)).

  • Binomials: Two terms (e.g., \(3x + 2\)).

  • Trinomials: Three terms (e.g., \(x^{2} + 4x + 4\)).
Polynomials are widely used in algebra for various operations such as addition, subtraction, multiplication, and division. By mastering the basics, you鈥檒l find it simpler to handle more complex polynomials.
Coefficients
Coefficients are the numerical values that multiply the variables in a polynomial.
In the polynomial \(4a^{3} - 4a^{2} - 4\), the coefficients are 4, -4, and -4 respectively. Coefficients are crucial as they determine the scale or magnitude of each term.
When adding polynomials, you only combine the coefficients of like terms. Here's how it works:
  • Identify the like terms in both polynomials.

  • Add or subtract their coefficients.

  • Keep the variable and exponent the same while you update the coefficient.
For example, in the given problem, to combine \(4a^{3}\) and \(6a^{3}\), we add their coefficients: 4 + 6, resulting in a new term \(10a^{3}\). Mastering coefficients will greatly enhance your understanding and ability to perform polynomial operations efficiently.

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Most popular questions from this chapter

Perform each division. $$ \frac{3 t^{4}+5 t^{3}-8 t^{2}-13 t+2}{t^{2}-5} $$

Match each expression in Column I with the equivalent expression in Column II. Choices in Column II may be used once, more than once, or not at all. An Exercise \(17, x \neq 0 .\). I (a) \(-2^{-4}\) (b) \((-2)^{-4}\) (c) \(2^{-4}\) (d) \(\frac{1}{2^{-4}}\) (e) \(\frac{1}{-2^{-4}}\) (f) \(\frac{1}{(-2)^{-4}}\) II \(\mathbf{A} .8\) \(\mathbf{B} .16\) \(\mathbf{C} .-\frac{1}{16}\) \(\mathbf{D} .-8\) \(\mathbf{E} .-16\) \(\mathbf{F} . \frac{1}{16}\)

Perform each division using the "long division" process. $$ \frac{5 z^{3}-z^{2}+10 z+2}{z+2} $$

To understand how the special product \((a+b)^{2}=a^{2}+2 a b+b^{2}\) can be applied to a purely numerical problem. The number 35 can be written as \(30+5 .\) Therefore, \(35^{2}=(30+5)^{2} .\) Use the special product for squaring a binomial with \(a=30\) and \(b=5\) to write an expression for \((30+5)^{2} .\) Do not simplify at this time.

Perform each division. $$ \left(x^{3}+2 x^{2}-3\right) \div(x-1) $$
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