Question

Simplify and write the resulting polynomial in descending order of degree. $$-7 x+5 x^{4}-2+3 x^{4}+7 x+10-3 x^{2}$$

Short answer

Question: Simplify the polynomial -7x + 5x^4 - 3x^2 + 7x - 2 + 3x^4 + 10 and arrange it in descending order of degree. Answer: The simplified polynomial is 8x^4 - 3x^2 + 8.

Step-by-step solution

Identify like terms

Identify the like terms in the given polynomial, which have the same exponent of the variable x. $$-7 x, 7 x$$ $$5 x^{4}, 3 x^{4}$$ $$-2, 10$$ $$-3 x^2$$

Combine the like terms

Combine the like terms by adding or subtracting the numerical coefficients of the like terms. $$ (-7x + 7x) + (5x^4 + 3x^4) + (-2 + 10) + (-3x^2) $$

Simplify the terms

Simplify the terms within the parentheses. $$ (0x) + (8x^4) + (8) + (-3x^2) $$

Rearrange the polynomial in descending order of degree and remove any term with omitted variables

Rearrange the terms of the simplified polynomial in descending order of degree. Since "0x" has no effect on the polynomial, we can remove it from our result. $$8x^4 - 3x^2 + 8$$ The simplified polynomial in descending order of degree is: $$8x^4 - 3x^2 + 8$$

Key concepts

Understanding Like Terms

When simplifying polynomials, one of the crucial steps is identifying and combining like terms. So, what are like terms? Simply put, like terms are terms that have the exact same variable raised to the same power. For instance, in the polynomial expression \(-7x + 5x^4 - 2 + 3x^4 + 7x + 10 - 3x^2\), we identify like terms by observing their exponents.
Here are the groups of like terms:

• \(-7x\) and \(7x\) are like terms because they both contain \(x\) raised to the first power.
• \(5x^4\) and \(3x^4\) are like terms as they both have \(x\) raised to the fourth power.
• \(-2\) and \(10\) are like terms as they are constant terms with no variables involved.
• \(-3x^2\) stands alone here as it does not share the same power of \(x\) with any other term.

Combining like terms means simply adding or subtracting their coefficients while keeping the variable part unchanged.

Rearranging in Descending Order

After simplifying polynomials by combining like terms, the next step is arranging the polynomial in descending order. Descending order refers to writing terms starting from the highest power of the variable down to the lowest. This approach makes interpreting and handling polynomials much easier.
For the given polynomial, after combining like terms, we have:

• \(0x\) which simplifies and disappears since it has no value.
• \(8x^4\)
• \(-3x^2\)
• \(+8\)

Writing this in descending order of degree means we start with the term having the highest degree, which is \(8x^4\), followed by \(-3x^2\), and finally the constant term \(8\). Thus, the polynomial rearranged in descending order is \(8x^4 - 3x^2 + 8\). This organization helps in easily identifying the nature and complexity of the polynomial at a glance.

Understanding Polynomial Degrees

A polynomial's degree is determined by the highest power (exponent) of the variable in the polynomial. The degree indicates the polynomial's highest level of complexity and plays a crucial role in graphing and solving polynomials.
In the polynomial \(8x^4 - 3x^2 + 8\), we notice:

• The term \(8x^4\) has the highest power of \(4\), therefore, the degree of this polynomial is \(4\).
• The significance of the degree is crucial when considering the polynomial's end behavior and its roots.
• Each term's degree determines its individual contribution to the overall polynomial.

Knowing the polynomial's degree helps in forecasting its graph's appearance, such as the number of turning points and crossing points along the axes. Essentially, understanding degrees is a fundamental part of working with and fully comprehending polynomials.