Simplify The Polynomial Expression:${ 8p^7 - 5p^5 - 2p^3 + P + 9 }$

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Polynomial expressions are a fundamental concept in algebra, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying polynomial expressions, using the given expression 8p7−5p5−2p3+p+98p^7 - 5p^5 - 2p^3 + p + 9 as an example.

Understanding Polynomial Expressions

A polynomial expression is a mathematical expression that consists of variables and coefficients combined using addition, subtraction, and multiplication. The variables in a polynomial expression can be raised to various powers, and the coefficients can be any real number. Polynomial expressions can be written in a variety of forms, including standard form, factored form, and expanded form.

Standard Form

The standard form of a polynomial expression is written with the terms in descending order of their powers. For example, the expression 8p7−5p5−2p3+p+98p^7 - 5p^5 - 2p^3 + p + 9 is already in standard form.

Factored Form

The factored form of a polynomial expression is written as a product of simpler expressions, called factors. For example, the expression 2p(p2+3p−4)2p(p^2 + 3p - 4) is in factored form.

Expanded Form

The expanded form of a polynomial expression is written with all the terms multiplied out. For example, the expression (p+2)(p−3)(p + 2)(p - 3) is in expanded form.

Simplifying Polynomial Expressions

Simplifying a polynomial expression involves combining like terms and eliminating any unnecessary terms. To simplify a polynomial expression, follow these steps:

  1. Combine like terms: Combine any terms that have the same variable and exponent. For example, in the expression 8p7−5p5−2p3+p+98p^7 - 5p^5 - 2p^3 + p + 9, the terms 8p78p^7 and −5p5-5p^5 are like terms, as are the terms −2p3-2p^3 and pp.
  2. Eliminate unnecessary terms: Eliminate any terms that are equal to zero. For example, in the expression 8p7−5p5−2p3+p+98p^7 - 5p^5 - 2p^3 + p + 9, the term 99 is not necessary, as it does not affect the value of the expression.
  3. Simplify the expression: Once you have combined like terms and eliminated unnecessary terms, simplify the expression by writing it in standard form.

Simplifying the Given Expression

Let's apply the steps above to simplify the given expression 8p7−5p5−2p3+p+98p^7 - 5p^5 - 2p^3 + p + 9.

Step 1: Combine Like Terms

The terms 8p78p^7 and −5p5-5p^5 are like terms, as are the terms −2p3-2p^3 and pp. We can combine these terms as follows:

  • 8p7−5p5=(8−5)p7=3p78p^7 - 5p^5 = (8 - 5)p^7 = 3p^7
  • −2p3+p=(−2+1)p3=−p3-2p^3 + p = (-2 + 1)p^3 = -p^3

So, the expression becomes 3p7−5p5−p3+93p^7 - 5p^5 - p^3 + 9.

Step 2: Eliminate Unnecessary Terms

The term 99 is not necessary, as it does not affect the value of the expression. We can eliminate this term by setting it equal to zero.

Step 3: Simplify the Expression

Now that we have combined like terms and eliminated unnecessary terms, we can simplify the expression by writing it in standard form.

The final simplified expression is 3p7−5p5−p33p^7 - 5p^5 - p^3.

Conclusion

Simplifying polynomial expressions is an essential skill for any math enthusiast. By following the steps outlined above, you can simplify even the most complex polynomial expressions. Remember to combine like terms, eliminate unnecessary terms, and simplify the expression to write it in standard form.

Example Problems

  • Simplify the expression 2p4−3p2+4p−52p^4 - 3p^2 + 4p - 5
  • Simplify the expression p3+2p2−3p+1p^3 + 2p^2 - 3p + 1
  • Simplify the expression 3p5−2p3+p2−4p+13p^5 - 2p^3 + p^2 - 4p + 1

Practice Problems

  • Simplify the expression p4−2p2+3p−1p^4 - 2p^2 + 3p - 1
  • Simplify the expression 2p3+3p2−4p+12p^3 + 3p^2 - 4p + 1
  • Simplify the expression p5−2p3+3p2−4p+1p^5 - 2p^3 + 3p^2 - 4p + 1

Glossary

  • Polynomial expression: A mathematical expression that consists of variables and coefficients combined using addition, subtraction, and multiplication.
  • Like terms: Terms that have the same variable and exponent.
  • Standard form: The form of a polynomial expression with the terms in descending order of their powers.
  • Factored form: The form of a polynomial expression as a product of simpler expressions, called factors.
  • Expanded form: The form of a polynomial expression with all the terms multiplied out.

References

  • "Algebra" by Michael Artin
  • "Polynomial Expressions" by Math Open Reference
  • "Simplifying Polynomial Expressions" by Khan Academy

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In our previous article, we explored the process of simplifying polynomial expressions. However, we know that practice makes perfect, and there's no better way to learn than by asking questions and getting answers. In this article, we'll address some of the most frequently asked questions about simplifying polynomial expressions.

Q: What is a polynomial expression?

A polynomial expression is a mathematical expression that consists of variables and coefficients combined using addition, subtraction, and multiplication. The variables in a polynomial expression can be raised to various powers, and the coefficients can be any real number.

Q: How do I simplify a polynomial expression?

To simplify a polynomial expression, follow these steps:

  1. Combine like terms: Combine any terms that have the same variable and exponent.
  2. Eliminate unnecessary terms: Eliminate any terms that are equal to zero.
  3. Simplify the expression: Once you have combined like terms and eliminated unnecessary terms, simplify the expression by writing it in standard form.

Q: What are like terms?

Like terms are terms that have the same variable and exponent. For example, in the expression 8p7−5p5−2p3+p+98p^7 - 5p^5 - 2p^3 + p + 9, the terms 8p78p^7 and −5p5-5p^5 are like terms, as are the terms −2p3-2p^3 and pp.

Q: How do I combine like terms?

To combine like terms, add or subtract the coefficients of the terms. For example, in the expression 8p7−5p5−2p3+p+98p^7 - 5p^5 - 2p^3 + p + 9, the terms 8p78p^7 and −5p5-5p^5 can be combined as follows:

  • 8p7−5p5=(8−5)p7=3p78p^7 - 5p^5 = (8 - 5)p^7 = 3p^7

Q: What is standard form?

Standard form is the form of a polynomial expression with the terms in descending order of their powers. For example, the expression 8p7−5p5−2p3+p+98p^7 - 5p^5 - 2p^3 + p + 9 is already in standard form.

Q: How do I eliminate unnecessary terms?

To eliminate unnecessary terms, set any terms that are equal to zero equal to zero. For example, in the expression 8p7−5p5−2p3+p+98p^7 - 5p^5 - 2p^3 + p + 9, the term 99 is not necessary, as it does not affect the value of the expression.

Q: Can I simplify a polynomial expression with variables of different powers?

Yes, you can simplify a polynomial expression with variables of different powers. For example, in the expression p3+2p2−3p+1p^3 + 2p^2 - 3p + 1, the terms p3p^3 and 2p22p^2 are like terms, as are the terms −3p-3p and 11.

Q: How do I simplify a polynomial expression with fractions?

To simplify a polynomial expression with fractions, multiply the numerator and denominator of each fraction by the least common multiple (LCM) of the denominators. For example, in the expression 12p3+34p2−13p+12\frac{1}{2}p^3 + \frac{3}{4}p^2 - \frac{1}{3}p + \frac{1}{2}, the LCM of the denominators is 1212. Multiply each fraction by 1212 to get:

  • 12p3=612p3\frac{1}{2}p^3 = \frac{6}{12}p^3
  • 34p2=912p2\frac{3}{4}p^2 = \frac{9}{12}p^2
  • −13p=−412p-\frac{1}{3}p = -\frac{4}{12}p
  • 12=612\frac{1}{2} = \frac{6}{12}

Now, combine like terms:

  • 612p3+912p2−412p+612=1712p3+912p2−412p+612\frac{6}{12}p^3 + \frac{9}{12}p^2 - \frac{4}{12}p + \frac{6}{12} = \frac{17}{12}p^3 + \frac{9}{12}p^2 - \frac{4}{12}p + \frac{6}{12}

Q: Can I simplify a polynomial expression with negative coefficients?

Yes, you can simplify a polynomial expression with negative coefficients. For example, in the expression −2p3+3p2−4p+1-2p^3 + 3p^2 - 4p + 1, the terms −2p3-2p^3 and 3p23p^2 are like terms, as are the terms −4p-4p and 11.

Q: How do I simplify a polynomial expression with exponents?

To simplify a polynomial expression with exponents, use the rules of exponents to combine like terms. For example, in the expression p3+2p2−3p+1p^3 + 2p^2 - 3p + 1, the terms p3p^3 and 2p22p^2 are like terms, as are the terms −3p-3p and 11.

Q: Can I simplify a polynomial expression with radicals?

Yes, you can simplify a polynomial expression with radicals. For example, in the expression 2p3+3p2−4p+1\sqrt{2}p^3 + \sqrt{3}p^2 - \sqrt{4}p + \sqrt{1}, the terms 2p3\sqrt{2}p^3 and 3p2\sqrt{3}p^2 are like terms, as are the terms −4p-\sqrt{4}p and 1\sqrt{1}.

Q: How do I simplify a polynomial expression with absolute values?

To simplify a polynomial expression with absolute values, use the definition of absolute value to rewrite the expression. For example, in the expression ∣p3+2p2−3p+1∣|p^3 + 2p^2 - 3p + 1|, the absolute value can be rewritten as:

  • ∣p3+2p2−3p+1∣=p3+2p2−3p+1|p^3 + 2p^2 - 3p + 1| = p^3 + 2p^2 - 3p + 1 if p3+2p2−3p+1≥0p^3 + 2p^2 - 3p + 1 \geq 0
  • ∣p3+2p2−3p+1∣=−(p3+2p2−3p+1)|p^3 + 2p^2 - 3p + 1| = -(p^3 + 2p^2 - 3p + 1) if p3+2p2−3p+1<0p^3 + 2p^2 - 3p + 1 < 0

Q: Can I simplify a polynomial expression with complex numbers?

Yes, you can simplify a polynomial expression with complex numbers. For example, in the expression p3+2p2−3p+1+2ip^3 + 2p^2 - 3p + 1 + 2i, the terms p3p^3 and 2p22p^2 are like terms, as are the terms −3p-3p and 11.

Q: How do I simplify a polynomial expression with trigonometric functions?

To simplify a polynomial expression with trigonometric functions, use the definitions of the trigonometric functions to rewrite the expression. For example, in the expression sin⁡(p3+2p2−3p+1)\sin(p^3 + 2p^2 - 3p + 1), the sine function can be rewritten as:

  • sin⁡(p3+2p2−3p+1)=sin⁡(p3)cos⁡(2p2−3p+1)+cos⁡(p3)sin⁡(2p2−3p+1)\sin(p^3 + 2p^2 - 3p + 1) = \sin(p^3) \cos(2p^2 - 3p + 1) + \cos(p^3) \sin(2p^2 - 3p + 1)

Q: Can I simplify a polynomial expression with logarithmic functions?

Yes, you can simplify a polynomial expression with logarithmic functions. For example, in the expression log⁡(p3+2p2−3p+1)\log(p^3 + 2p^2 - 3p + 1), the logarithmic function can be rewritten as:

  • log⁡(p3+2p2−3p+1)=log⁡(p3)+log⁡(2p2−3p+1)\log(p^3 + 2p^2 - 3p + 1) = \log(p^3) + \log(2p^2 - 3p + 1)

Q: How do I simplify a polynomial expression with exponential functions?

To simplify a polynomial expression with exponential functions, use the definition of the exponential function to rewrite the expression. For example, in the expression ep3+2p2−3p+1e^{p^3 + 2p^2 - 3p + 1}, the exponential function can be rewritten as:

  • ep3+2p2−3p+1=ep3e2p2−3p+1e^{p^3 + 2p^2 - 3p + 1} = e^{p^3} e^{2p^2 - 3p + 1}

Q: Can I simplify a polynomial expression with inverse functions?

Yes, you can simplify a polynomial expression with inverse functions. For example, in the expression 1p3+2p2−3p+1\frac{1}{p^3 + 2p^2 - 3p + 1}, the inverse function can be rewritten as:

  • 1p3+2p2−3p+1=1p312p2−3p+1\frac{1}{p^3 + 2p^2 - 3p + 1} = \frac{1}{p^3} \frac{1}{2p^2 - 3p + 1}

Q: How do I simplify a polynomial expression with rational functions?

To simplify a polynomial expression with rational functions, use the definition of the rational function to rewrite the expression. For example, in the expression $\frac{