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Answer :
To determine if a polynomial is in standard form, we must check whether its terms are arranged in descending order according to the degree (exponent) of the variable. In other words, the exponents should decrease from left to right.
Let's analyze each option:
1. The first polynomial is
[tex]$$2x^4 + 6 + 24x^5.$$[/tex]
The exponents here are [tex]$4$[/tex], [tex]$0$[/tex], and [tex]$5$[/tex], respectively. Since [tex]$5$[/tex] (from the last term) is greater than [tex]$4$[/tex] and not in descending order, this polynomial is not in standard form.
2. The second polynomial is
[tex]$$6x^2 - 9x^3 + 12x^4.$$[/tex]
The exponents are [tex]$2$[/tex], [tex]$3$[/tex], and [tex]$4$[/tex]. They are arranged in ascending order rather than descending order. Therefore, this is not in standard form.
3. The third polynomial is
[tex]$$19x + 6x^2 + 2.$$[/tex]
The exponents are [tex]$1$[/tex], [tex]$2$[/tex], and [tex]$0$[/tex] respectively. Since the term with exponent [tex]$2$[/tex] does not come first, this polynomial is not written 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]. Since [tex]$9 > 4 > 0$[/tex], the terms are arranged in descending order.
Since the fourth option is the only one with its terms arranged in descending order by their exponents, the polynomial in standard form is
[tex]$$23x^9 - 12x^4 + 19.$$[/tex]
Let's analyze each option:
1. The first polynomial is
[tex]$$2x^4 + 6 + 24x^5.$$[/tex]
The exponents here are [tex]$4$[/tex], [tex]$0$[/tex], and [tex]$5$[/tex], respectively. Since [tex]$5$[/tex] (from the last term) is greater than [tex]$4$[/tex] and not in descending order, this polynomial is not in standard form.
2. The second polynomial is
[tex]$$6x^2 - 9x^3 + 12x^4.$$[/tex]
The exponents are [tex]$2$[/tex], [tex]$3$[/tex], and [tex]$4$[/tex]. They are arranged in ascending order rather than descending order. Therefore, this is not in standard form.
3. The third polynomial is
[tex]$$19x + 6x^2 + 2.$$[/tex]
The exponents are [tex]$1$[/tex], [tex]$2$[/tex], and [tex]$0$[/tex] respectively. Since the term with exponent [tex]$2$[/tex] does not come first, this polynomial is not written 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]. Since [tex]$9 > 4 > 0$[/tex], the terms are arranged in descending order.
Since the fourth option is the only one with its terms arranged in descending order by their exponents, the polynomial in standard form is
[tex]$$23x^9 - 12x^4 + 19.$$[/tex]
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