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Answer :
A polynomial is in standard form when its terms are arranged in descending order by the exponent of [tex]$x$[/tex]. Let’s examine each option:
1. The first polynomial is
[tex]$$2x^4 + 6 + 24x^5.$$[/tex]
The exponents of [tex]$x$[/tex] in the terms are [tex]$4$[/tex], [tex]$0$[/tex], and [tex]$5$[/tex]. Since [tex]$0$[/tex] is not greater than [tex]$5$[/tex], the order is not strictly descending.
2. The second polynomial is
[tex]$$6x^2 - 9x^3 + 12x^4.$$[/tex]
The exponents here are [tex]$2$[/tex], [tex]$3$[/tex], and [tex]$4$[/tex], which are in ascending (increasing) order rather than descending order.
3. The third polynomial is
[tex]$$19x + 6x^2 + 2.$$[/tex]
The exponents are [tex]$1$[/tex], [tex]$2$[/tex], and [tex]$0$[/tex]. This sequence is also not in descending order.
4. The fourth polynomial is
[tex]$$23x^9 - 12x^4 + 19.$$[/tex]
The exponents are [tex]$9$[/tex], [tex]$4$[/tex], and [tex]$0$[/tex], which are in descending order since [tex]$9 > 4 > 0$[/tex].
Thus, the polynomial in standard form is the fourth one.
The final answer is: Option 4.
1. The first polynomial is
[tex]$$2x^4 + 6 + 24x^5.$$[/tex]
The exponents of [tex]$x$[/tex] in the terms are [tex]$4$[/tex], [tex]$0$[/tex], and [tex]$5$[/tex]. Since [tex]$0$[/tex] is not greater than [tex]$5$[/tex], the order is not strictly descending.
2. The second polynomial is
[tex]$$6x^2 - 9x^3 + 12x^4.$$[/tex]
The exponents here are [tex]$2$[/tex], [tex]$3$[/tex], and [tex]$4$[/tex], which are in ascending (increasing) order rather than descending order.
3. The third polynomial is
[tex]$$19x + 6x^2 + 2.$$[/tex]
The exponents are [tex]$1$[/tex], [tex]$2$[/tex], and [tex]$0$[/tex]. This sequence is also not in descending order.
4. The fourth polynomial is
[tex]$$23x^9 - 12x^4 + 19.$$[/tex]
The exponents are [tex]$9$[/tex], [tex]$4$[/tex], and [tex]$0$[/tex], which are in descending order since [tex]$9 > 4 > 0$[/tex].
Thus, the polynomial in standard form is the fourth one.
The final answer is: Option 4.
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