Answer :
To determine if a polynomial is written in standard form, we need to check whether its terms are arranged in descending order by the exponents of [tex]$x$[/tex].
Let's examine each option:
1. For the polynomial
[tex]$$2x^4 + 6 + 24x^5,$$[/tex]
the exponents are [tex]$4$[/tex], [tex]$0$[/tex], and [tex]$5$[/tex], respectively. For standard form, the terms should be arranged from highest to lowest degree. The correct descending order should be [tex]$x^5$[/tex], then [tex]$x^4$[/tex], and finally the constant term. Since the given order does not follow this, the polynomial is not in standard form.
2. For the polynomial
[tex]$$6x^2 - 9x^3 + 12x^4,$$[/tex]
the exponents are [tex]$2$[/tex], [tex]$3$[/tex], and [tex]$4$[/tex], respectively. In standard form, the highest exponent ([tex]$4$[/tex]) should come first, followed by [tex]$3$[/tex], and then [tex]$2$[/tex]. The given arrangement does not follow this order, so this polynomial is not in standard form.
3. For the polynomial
[tex]$$19x + 6x^2 + 2,$$[/tex]
the exponents are [tex]$1$[/tex], [tex]$2$[/tex], and [tex]$0$[/tex], respectively. The correct descending order should be [tex]$x^2$[/tex], then [tex]$x$[/tex], then the constant. Since the terms here are not in descending order, the polynomial is not in standard form.
4. For the polynomial
[tex]$$23x^9 - 12x^4 + 19,$$[/tex]
the exponents are [tex]$9$[/tex], [tex]$4$[/tex], and [tex]$0$[/tex]. This order is already in descending order. Therefore, this polynomial is in standard form.
Thus, the polynomial in standard form is the fourth one:
[tex]$$23x^9 - 12x^4 + 19.$$[/tex]
Let's examine each option:
1. For the polynomial
[tex]$$2x^4 + 6 + 24x^5,$$[/tex]
the exponents are [tex]$4$[/tex], [tex]$0$[/tex], and [tex]$5$[/tex], respectively. For standard form, the terms should be arranged from highest to lowest degree. The correct descending order should be [tex]$x^5$[/tex], then [tex]$x^4$[/tex], and finally the constant term. Since the given order does not follow this, the polynomial is not in standard form.
2. For the polynomial
[tex]$$6x^2 - 9x^3 + 12x^4,$$[/tex]
the exponents are [tex]$2$[/tex], [tex]$3$[/tex], and [tex]$4$[/tex], respectively. In standard form, the highest exponent ([tex]$4$[/tex]) should come first, followed by [tex]$3$[/tex], and then [tex]$2$[/tex]. The given arrangement does not follow this order, so this polynomial is not in standard form.
3. For the polynomial
[tex]$$19x + 6x^2 + 2,$$[/tex]
the exponents are [tex]$1$[/tex], [tex]$2$[/tex], and [tex]$0$[/tex], respectively. The correct descending order should be [tex]$x^2$[/tex], then [tex]$x$[/tex], then the constant. Since the terms here are not in descending order, the polynomial is not in standard form.
4. For the polynomial
[tex]$$23x^9 - 12x^4 + 19,$$[/tex]
the exponents are [tex]$9$[/tex], [tex]$4$[/tex], and [tex]$0$[/tex]. This order is already in descending order. Therefore, this polynomial is in standard form.
Thus, the polynomial in standard form is the fourth one:
[tex]$$23x^9 - 12x^4 + 19.$$[/tex]
