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Chapter 5: Problem 17

Add $$ \left(9 x^{8}-7 x^{4}+2 x^{2}+5\right)+\left(8 x^{7}+4 x^{4}-2 x\right) $$

Short Answer

Expert verified
9x^{8} + 8x^{7} - 3x^{4} + 2x^{2} - 2x + 5

Step by step solution

01

Identify Like Terms

Examine both polynomials to identify like terms, which are terms that have the same variable raised to the same power.
02

Combine Like Terms

Add the coefficients of like terms together. If a term does not have a like term in the other polynomial, it remains unchanged.
03

Write the Final Polynomial

Combine all the resulting terms into a single polynomial, ordered by decreasing powers of x.

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Key Concepts

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

Like Terms
When dealing with polynomials, it's important to understand the concept of 'like terms'. Like terms are terms that have the same variable raised to the same power. For example, in the expression 9x^8 + 4x^4, the terms 9x^8 and 4x^4 are not like terms because the exponents of x are different. However, in the expression 9x^8 - 7x^4 + 4x^4, the terms -7x^4 and 4x^4 are like terms because they both have x raised to the fourth power. Identifying like terms is the first step in adding polynomials, as it allows you to combine them effectively.
Coefficients
Coefficients are the numerical parts of terms in polynomials. They are the numbers that multiply the variables. For instance, in the term 9x^8, the coefficient is 9. Similarly, in the term -7x^4, the coefficient is -7. When combining like terms, you only add or subtract the coefficients, while the variable parts remain unchanged. For example, if you are adding -7x^4 and 4x^4, you would add their coefficients (-7 and 4) to get -3. Thus, -7x^4 + 4x^4 = -3x^4. This process simplifies the polynomial and makes it easier to work with.
Combining Polynomials
Combining polynomials involves adding or subtracting like terms. Here鈥檚 a clear example using the original exercise. Consider the two polynomials: (9x^8 - 7x^4 + 2x^2 + 5) and (8x^7 + 4x^4 - 2x). First, identify the like terms: - 9x^8 has no like term - -7x^4 and 4x^4 are like terms - 2x^2 has no like term - 5 has no like term - 8x^7 has no like term - -2x has no like term. Next, add or subtract the coefficients of the like terms: Combining -7x^4 and 4x^4, you get -3x^4. Finally, write the resulting polynomial, ordering the terms by decreasing powers of x: 9x^8 + 8x^7 - 3x^4 + 2x^2 - 2x + 5. This is the final polynomial.

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Most popular questions from this chapter

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