Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Discover a wealth of knowledge from experts across different disciplines on our comprehensive Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
Alright, let's identify which of the given expressions are polynomials by carefully examining each term in the expressions. A polynomial is an expression involving only non-negative integer exponents of the variable.
### Expression (a)
[tex]\[3 + 2x - 5x^2 - 4x^3\][/tex]
- The constant term [tex]\(3\)[/tex] is a polynomial term.
- The term [tex]\(2x\)[/tex] has an exponent of 1, which is a non-negative integer.
- The term [tex]\(-5x^2\)[/tex] has an exponent of 2, which is a non-negative integer.
- The term [tex]\(-4x^3\)[/tex] has an exponent of 3, which is a non-negative integer.
All terms in the expression have non-negative integer exponents. Therefore, expression (a) is a polynomial.
### Expression (b)
[tex]\[\sqrt{4} x^3 + 7x^2 - 8x + 9\][/tex]
- The term [tex]\(\sqrt{4} x^3\)[/tex] simplifies since [tex]\(\sqrt{4} = 2\)[/tex], yielding [tex]\(2x^3\)[/tex], which has an exponent of 3. This is a polynomial term.
- The term [tex]\(7x^2\)[/tex] has an exponent of 2, which is a non-negative integer.
- The term [tex]\(-8x\)[/tex] has an exponent of 1, which is a non-negative integer.
- The constant term [tex]\(9\)[/tex] is a polynomial term.
All terms in the expression have non-negative integer exponents. Therefore, expression (b) is a polynomial.
### Expression (c)
[tex]\[3x^4 + 9x^3 - 7\sqrt{x} + 8\][/tex]
- The term [tex]\(3x^4\)[/tex] has an exponent of 4, which is a non-negative integer.
- The term [tex]\(9x^3\)[/tex] has an exponent of 3, which is a non-negative integer.
- The term [tex]\(-7\sqrt{x}\)[/tex] or [tex]\(-7x^{1/2}\)[/tex] has an exponent of [tex]\(\frac{1}{2}\)[/tex], which is not an integer.
Because the exponent [tex]\(\frac{1}{2}\)[/tex] in the term [tex]\(-7\sqrt{x}\)[/tex] is not a non-negative integer, expression (c) is not a polynomial.
### Expression (d)
[tex]\[4x^3 - 10x^{-1} + 1\][/tex]
- The term [tex]\(4x^3\)[/tex] has an exponent of 3, which is a non-negative integer.
- The term [tex]\(-10x^{-1}\)[/tex] has an exponent of [tex]\(-1\)[/tex], which is negative.
Because the exponent [tex]\(-1\)[/tex] in the term [tex]\(-10x^{-1}\)[/tex] is negative, expression (d) is not a polynomial.
### Summary
- Expression (a) is a polynomial.
- Expression (b) is a polynomial.
- Expression (c) is not a polynomial because it contains the term [tex]\(\sqrt{x}\)[/tex], with a non-integer exponent of [tex]\(\frac{1}{2}\)[/tex].
- Expression (d) is not a polynomial because it contains the term [tex]\(x^{-1}\)[/tex], with a negative exponent.
### Expression (a)
[tex]\[3 + 2x - 5x^2 - 4x^3\][/tex]
- The constant term [tex]\(3\)[/tex] is a polynomial term.
- The term [tex]\(2x\)[/tex] has an exponent of 1, which is a non-negative integer.
- The term [tex]\(-5x^2\)[/tex] has an exponent of 2, which is a non-negative integer.
- The term [tex]\(-4x^3\)[/tex] has an exponent of 3, which is a non-negative integer.
All terms in the expression have non-negative integer exponents. Therefore, expression (a) is a polynomial.
### Expression (b)
[tex]\[\sqrt{4} x^3 + 7x^2 - 8x + 9\][/tex]
- The term [tex]\(\sqrt{4} x^3\)[/tex] simplifies since [tex]\(\sqrt{4} = 2\)[/tex], yielding [tex]\(2x^3\)[/tex], which has an exponent of 3. This is a polynomial term.
- The term [tex]\(7x^2\)[/tex] has an exponent of 2, which is a non-negative integer.
- The term [tex]\(-8x\)[/tex] has an exponent of 1, which is a non-negative integer.
- The constant term [tex]\(9\)[/tex] is a polynomial term.
All terms in the expression have non-negative integer exponents. Therefore, expression (b) is a polynomial.
### Expression (c)
[tex]\[3x^4 + 9x^3 - 7\sqrt{x} + 8\][/tex]
- The term [tex]\(3x^4\)[/tex] has an exponent of 4, which is a non-negative integer.
- The term [tex]\(9x^3\)[/tex] has an exponent of 3, which is a non-negative integer.
- The term [tex]\(-7\sqrt{x}\)[/tex] or [tex]\(-7x^{1/2}\)[/tex] has an exponent of [tex]\(\frac{1}{2}\)[/tex], which is not an integer.
Because the exponent [tex]\(\frac{1}{2}\)[/tex] in the term [tex]\(-7\sqrt{x}\)[/tex] is not a non-negative integer, expression (c) is not a polynomial.
### Expression (d)
[tex]\[4x^3 - 10x^{-1} + 1\][/tex]
- The term [tex]\(4x^3\)[/tex] has an exponent of 3, which is a non-negative integer.
- The term [tex]\(-10x^{-1}\)[/tex] has an exponent of [tex]\(-1\)[/tex], which is negative.
Because the exponent [tex]\(-1\)[/tex] in the term [tex]\(-10x^{-1}\)[/tex] is negative, expression (d) is not a polynomial.
### Summary
- Expression (a) is a polynomial.
- Expression (b) is a polynomial.
- Expression (c) is not a polynomial because it contains the term [tex]\(\sqrt{x}\)[/tex], with a non-integer exponent of [tex]\(\frac{1}{2}\)[/tex].
- Expression (d) is not a polynomial because it contains the term [tex]\(x^{-1}\)[/tex], with a negative exponent.
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
